Talk:Fibonacci sequence/Archive 2

This is an archive of past discussions about Fibonacci sequence. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

Archive 2

Archive 3

Archive 4

Gopala / Hemachandra

I've removed the clause from the introduction that says that the numbers are also called the "Gopala-Hemachandra numbers". The page already mentions that Fibonacci was anticipated by Gopala and Hemachandra, and I find no evidence that the numbers are actually called the "Gopala-Hemachandra numbers".

I'm also going to redirect the Gopala-Hemachandra numbers article to this one, since the two phrases mean the same thing and that article contains nothing that isn't already in this one.

-- Dominus 14:16, 11 Nov 2004 (UTC)

Addendum: even the external research paper linked to from the Gopala-Hemachandra numbers page does not refer to the numbers as the "Gopala-Hemachandra numbers". It says "The numbers in the sequence are called Fibonacci numbers." The phrase "Gopala-Hemachandra numbers" does not appear in that paper.

-- Dominus 14:18, 11 Nov 2004 (UTC)

i agree that the internal reaseach paper linked to from the page definatley does not refer to the numbers as gopala-hemachandra numbers.i have checked twice over and it does not apper in the paper. —Preceding unsigned comment added by 213.1.35.46 (talk) 14:35, 25 September 2007 (UTC)

I note we still have a page Gopala-Hemachandra number (no s at the end) which is not a redirect. I've now redirected it to here. --Salix alba (talk) 14:53, 25 September 2007 (UTC)

Bartok

From the "Application" section: "It is commonly thought that the first movement of Béla Bartók's Music for Strings, Percussion, and Celesta was structured using Fibonacci numbers."

Well maybe it is commonly thought, but that doesn't mean it is true. Until someone can come up with an explanation on why that movement has 88 bars and not 89 as the Fibonacci sequence would suggest, I would like to see this part removed from the article. NguyenVanThoc 22:41, 30 November 2007 (UTC)

Formula

While these nifty bignum formulas are nice and all, I think it would be very nice to have the actual formula for calculating them. The math isn't that hard to do by hand, because of cancelling pieces. I think that the article needs it because it is the non recursive form of it.

See the Closed Form Expression, which translates to:

F\left(n\right)=({1 \over {\sqrt {5}}})(({1+{\sqrt {5}} \over 2})^{n}-({1-{\sqrt {5}} \over 2})^{n}))

Courtesy of Posamentier and Lehmann^[1] -Dagordon01 16:40, 1 December 2007 (UTC)

Formula of finding fibonacci number

lets say a = sqrt(5), then: F(n) = ((a+1)^(n+1)-(a-1))/(2^(n+1)*a)

See the discussion above about "Formula" and the reference to Posamentier and Lehmann^[2] -Dagordon01 16:52, 1 December 2007 (UTC)

Popular Culture

This section is VERY vague. Should some examples be given? —Preceding unsigned comment added by BrettxPW (talk • contribs) 20:52, 10 December 2007 (UTC)

Identity for doubling n

I added the actual identity for doubling n. I think the formula for F_{2n+k) needs a reference or something since I have never seen that before. The reference provided right below that does NOT contain that identity and indeed contains an identity that is completely wrong: F_2n=F_{n}^2+F_{n-1}^2. I believe that reference should be removed. I also don't see how it reduces to the F_2n formula when k = 0. Also, it should definitely not say for all integers k and n because it doesn't make sense if n<0 or k<3. (SlaterDeterminant (talk) 16:44, 3 January 2008 (UTC))

I fixed the formula that you added for F_2n. The F_2n+k formula looks fine to me - it is just a special case of Formula 47 from the MathWorld page. When k=0 you have F_k=0, F_k-1=1 and F_k-2=-1, so you get F_2n = 2F_n+1F_n - F_n² as expected. Gandalf61 (talk) 17:17, 3 January 2008 (UTC)

Proof by induction

Why haven’t you completed your Proof by induction of Binet’s formula? You’ve shown its true for 0 and 1.I think you now need to show that if it’s true for n and n+1 then it is also true for n+2,the dominoes topple, and you’ve proved it for all the natural numbers. I’ve just tried to do this on a bit of paper and I can’t.It certainly isn’t so obvious you can just leave it out! —Preceding unsigned comment added by 91.107.165.60 (talk) 21:23, 9 January 2008 (UTC)

Duh! I’ve just seen how to do it. It is pretty obvious but someone who can write Latex ought to put it up. —Preceding unsigned comment added by 91.107.165.60 (talk) 21:39, 9 January 2008 (UTC)

I’ve just tried to do it by cutting and pasting Lyx code but that doesn’t work. I get “parsing error”. I’m not going to learn all the bloody code, someone else will have to do it. Anyone who thinks for 5 minutes will see how the proof works anyway. It just annoys me it isn’t completed. Dave59 (talk) 23:01, 9 January 2008 (UTC)

let P(n) be the variable proposition $F_{n}={\frac {\phi ^{n}-\psi ^{n}}{\sqrt {5}}}$

P(n+1) is $F_{n+1}={\frac {\phi ^{n+1}-\psi ^{n+1}}{\sqrt {5}}}$

P(n+2) is $F_{n+2}={\frac {\phi ^{n+2}-\psi ^{n+2}}{\sqrt {5}}}$

Now $F_{n+2}=F_{n+1}+F_{n}$

Therefore $F_{n+2}={\frac {\phi ^{n}-\psi ^{n}}{\sqrt {5}}}+{\frac {\phi ^{n+1}-\psi ^{n+1}}{\sqrt {5}}}={\frac {\left(\phi ^{n}+\phi ^{n+1}\right)-\left(\psi ^{n}+\psi ^{n+1}\right)}{\sqrt {5}}}$

we have allready shown $\phi ^{n+2}=\phi ^{n+1}+\phi ^{n}\qquad and\qquad \psi ^{n+2}=\psi ^{n+1}+\psi ^{n}$

Therefore $F_{n+2}={\frac {\phi ^{n+2}-\psi ^{n+2}}{\sqrt {5}}}$

So $P(n)\qquad and\qquad P(n+1)\Rightarrow P(n+2)$

we have already shown P(0) and P(1) are true

Therfore by mathematical induction the proposition is proved for all natural numbers.(or for all the natural numbers plus zero if you want to be really pedantic)

Code a damn site harder than the maths

Dave. —Preceding unsigned comment added by 91.105.18.197 (talk) 12:42, 11 January 2008 (UTC)

This seems already explained in Fibonacci_number#Proof_by_induction.--Patrick (talk) 13:46, 11 January 2008 (UTC)

It probably says enough for a mathematician to understand the drift straight away. However it is not a formal proof by induction and I didn’t understand it the first time I read it. This just dots the i’s and crosses the t’s. This is pure maths and I feel we ought to be precise. I have used slightly different notation to the main article. I’m unfamiliar with Latex and this took me ages to do. I’m not even going to try to integrate it into the main article. It is probably true that most of the people who are going to read the article don’t need it but it might be useful for people who are just learning proof by induction and want to see a few examples. Dave59 (talk) 15:37, 11 January 2008 (UTC)

Fibonacci sequence

This article should be called Fibonacci sequence and not Fibonacci number. A Fibonacci number is meaningless out of the context of its sequence. If I asked you "what is 21?", nobody would say "the Fibonacci number after 13". But if I asked "what is 1, 1, 2, 3, 5, 8, 13, 21...?, I'd have a much greater chance of hearing "Fibonacci sequence". This article should be moved to Fibonacci sequence over the redirect, and Fibonacci number should redirect to Fibonacci sequence. TableManners^C·U·T 06:05, 18 January 2008 (UTC)

I was going to say it's commonly called "numbers" by everyone in the world, but then I looked at the interwiki links: bg, cs, eo, pt, ru - Numbers. ca, de, el, es, fr, it, scn, sk, tr, uk - Sequence. Still, I've mostly seen it as "numbers" in English - for example that's how it's called on the Integer Sequences site [1] and, for another example, Wolfram's Mathworld defines the Sequence [2] as "see Fibonacci Number". The Marriam-Webster dictionary of the English Language has the entry for numbers [3] but not sequence, while American Heritage Dictionary has both and essentially says "See Sequence" for Number: [4] and [5]. Doesn't look like there is an agreement. --Cubbi (talk) 12:23, 18 January 2008 (UTC)

Though I'm not sure it's supported by Cubbi's post, isn't "sequence" a alightly technical mathematician's way of putting it, and "numbers" what the man in the street would say? Fibonacci numbers are rather insignificant in professional math, but play a quite significant role in popular math, recreational math. I'm for keeping the article at "numbers".--Niels Ø (noe) (talk) 13:19, 18 January 2008 (UTC)

Conditions such as "if n is a Fibonacci number" naturally arise, independent of any overt connexion to the sequence, often enough that I disagree with TableManners: they are a meaningful set or class of numbers. —Tamfang (talk) 23:07, 22 January 2008 (UTC)

Just passing by, figured I'd share my 2 cents. Google results:

"Fibonacci sequence" (with quotes): ~186,000

"Fibonacci number" (with quotes): ~86,000

"Fibonacci numbers" (with quotes): ~216,000

Therefore, I propose a move to Fibonacci numbers with Fibonacci sequence and Fibonacci number redirecting to that title. FireCrotch (talk) 15:18, 28 January 2008 (UTC)

Current convention for Wikipedia articles on integer sequences is to name them xxx number or xxx sequence but never xxx numbers - see Category:Integer sequences for many examples. This follows Wikipedia:Naming conventions, which says "In general only create page titles that are in the singular, unless that noun is always in a plural form in English (such as scissors or trousers)". Gandalf61 (talk) 15:32, 28 January 2008 (UTC)

Thank you for pointing that out, Gandolf61. In that case, I suggest that it be renamed to "Fibonacci sequence". Now that I think about it, of course "Fibonacci numbers" is going to have more results - it includes all pages that contain "Fibonacci number" as well! FireCrotch (talk) 04:00, 29 January 2008 (UTC)

Why two separate "Pythagorean triple" sections?

"Pythagorean triples of Fibonacci numbers" is the subject of two separate sections of this article:

"Right triangles," and
"Pythagorean triples"

I'm not sure what might be the most parsimonious/harmonious way to do it, but wouldn't it be best to somehow merge these sections?
—Wikiscient— 11:01, 12 March 2008 (UTC)

I merged them. —David Eppstein (talk) 14:46, 12 March 2008 (UTC)

Bees - Got it backwards I think

Males only come from mated bees (how can a female introduce a male chromosome?) The logic in how it relates to the Fibonacci sequence is still the same, but I think male and female were switched in the logic. I have changed it, and if you find I am wrong (with references of course) feel free to undo my switch. I only found this link as a reference for now, maybe will come back later with more. --Billy Nair (talk) 16:39, 16 March 2008 (UTC)

I am pretty sure the same is with chickens, unfertalized eggs will be female, and only fertalized eggs have the chance to be male, I don't know if it is always male, but need a male to get a male. --Billy Nair (talk) 16:41, 16 March 2008 (UTC)

Sorry, but the article was right - male bees, also known as drones, develop from unfertilised eggs. See our drone (bee) article, which explains how the genetics works. I have changed the article back to how it was. Gandalf61 (talk) 17:35, 16 March 2008 (UTC)

Birds, I believe, have a system analogous to the XY of mammals but the other way around: a bird with matching chromosomes is male, one with differing sex chromosomes is female; but a haploid (unfertilized) egg won't develop at all. —Tamfang (talk) 22:22, 3 April 2008 (UTC)

Labels in the list of values

I just reverted a change by Virginia-American (talk · contribs) that replaced the list of values near the start of the article,

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, ...

by a list in which each value is labeled,

F₀ = 0, F₁ = 1, F₂ = 1, F₃ = 2, F₄ = 3, F₅ = 5,

F₆ = 8, F₇ = 13, F₈ = 21, F₉ = 34, F₁₀ = 55,

F₁₁ = 89, F₁₂ = 144, F₁₃ = 233, F₁₄ = 377, F₁₅ = 610,

F₁₆ = 987, F₁₇ = 1597, F₁₈ = 2584, F₁₉ = 4181, F₂₀ = 6765,

F₂₁ = 10946, F₂₂ = 17711, F₂₃ = 28657, F₂₄ = 46368, F₂₅ = 75025,

F₂₆ = 121393, ...

I think the labels make the list completely unreadable. But since this is a content disagreement rather than something more clear-cut, I thought I'd bring it here for further discussion, if there is any. —David Eppstein (talk) 15:40, 22 March 2008 (UTC)

Obviously I disagree. I got annoyed trying to verify some of the formulas and having to count to see what value of n corresponded to which fibonacci number. Actually, a table of some sort is probably the best way to display the first few values. Virginia-American (talk) 15:50, 22 March 2008 (UTC)

I made a table. I agree that the n is needed.--Patrick (talk) 16:13, 22 March 2008 (UTC)

Sorry, but I reverted the table. If you have TOC switched on the table runs down the left hand side of the TOC and looks just awful. And it makes the TOC appear in the middle of the lead section, for some reason. I wouldn't object to a table if (a) it is multi-column so it takes up less vertical space and (b) it can be arranged so as not to overlap the TOC. Gandalf61 (talk) 18:15, 22 March 2008 (UTC)

I made a table going horizontally instead of vertically. I think it looks better, but in order to work with possibly narrow browser windows I truncated the sequence earlier (21 terms). —David Eppstein (talk) 18:59, 22 March 2008 (UTC)

Yes, that looks better. Good job. Gandalf61 (talk) 21:16, 22 March 2008 (UTC)

Thanks for the table - great reference-ability on that. Bugtank (talk) 04:47, 6 April 2008 (UTC)

The most notable property of the Fibonacci sequence is the ratio converting to $\varphi$ = 1.6180339887... Would it be worth the space to put something like the below table in a section (not the lead)? There have been complaints that the article is complicated. The table is simple and could come before more complicated parts. PrimeHunter (talk) 21:51, 22 March 2008 (UTC)

Fibonacci numbers
n	F_n	Factorization	F_n / F_n-1	abs(F_n / F_n-1 − $\varphi$ )
0	0
1	1	1
2	1	1	1	0.6180339887
3	2	2	2	0.3819660113
4	3	3	1.5	0.1180339887
5	5	5	1.6666666667	0.0486326779
6	8	2³	1.6	0.0180339887
7	13	13	1.625	0.0069660112
8	21	3·7	1.6153846154	0.0026493733
9	34	2·17	1.6190476190	0.0010136302
10	55	5·11	1.6176470588	0.0003869299
11	89	89	1.6181818182	0.0001478294
12	144	2⁴·3²	1.6179775281	0.0000564606
13	233	233	1.6180555556	0.0000215668
14	377	13·29	1.6180257511	0.0000082376
15	610	2·5·61	1.6180371353	0.0000031465
16	987	3·7·47	1.6180327869	0.0000012018
17	1597	1597	1.6180344478	0.0000004590
18	2584	2³·17·19	1.6180338134	0.0000001753
19	4181	37·113	1.6180340557	0.0000000669
20	6765	3·5·11·41	1.6180339632	0.0000000255
21	10946	2·13·421	1.6180339985	0.0000000097
22	17711	89·199	1.6180339850	0.0000000037
23	28657	28657	1.6180339902	0.0000000014
24	46368	2⁵·3²·7·23	1.6180339882	0.0000000005
25	75025	5²·3001	1.6180339890	0.0000000002

PrimeHunter said "The most notable property of the Fibonacci sequence is the ratio converting to

\varphi

= 1.6180339887... ".

Not really. It doesn't matter what the starting values are, as long as a sequence has the same recurrence F_n+1 = F_n + F_n-1 as the Fibonacci sequence, the ratios converge to phi. I think the fact that the convergents of the continued fraction for phi are the ratios of consecutive F_n s is more notable. Virginia-American (talk) 17:35, 27 March 2008 (UTC)

I meant notable as in Wikipedia:Notability, meaning there are lots of sources about it. I think your property is mentioned relatively rarely. PrimeHunter (talk) 17:42, 27 March 2008 (UTC)

Error in proof

I fixed an error in the proof of the third identity. Paul August ☎ 05:44, 9 April 2008 (UTC)

Fibonacci Name

I have always known these specific numbers as the Fibonacci Sequence. I was surprised to find them named Fibonacci Number. Does anyone now if Fibonacci Number is the exact name? Or which name would be more recognizable? WebberTakito 02:44, 21 June 2008 (UTC)

They are both common. I currently get 208000 Google hits on "Fibonacci sequence" and 249000 on "Fibonacci numbers". PrimeHunter (talk) 03:18, 21 June 2008 (UTC)

AfD of popular culture article

This article links to Fibonacci numbers in popular culture. That article has been nominated for deletion. If such deleteion were to happen, this present article would be effected in two ways: (1) that link to be deleted; (2) an abbreviated popular culture section would probably get added to this article. I think the culture article is probably worth keeping but needs improvement, and probably best kept as a separate article. Here are some thoughts I put on the AfD page and on the culture article's talk page:

The topic is notable for this reason: allusions to the Fibonacci numbers in writing or speaking on virtually any subject are widely understood.

Now the fact is, many of the items now listed on this page are not sufficiently notable to be a topic in an encyclopedia. That shouldn't matter here, since it is not necessary for individual instances of a mode of allusion to be notable in order for them to illustrate that the allusion itself is notable. But I think priority should be given to examples that do illustrate that.

Michael Hardy (talk) 19:55, 24 June 2008 (UTC)

Plot

There are errors in this:

Patrick (talk) 16:16, 22 March 2008 (UTC)

Errors fixed. Thankyousarindam7 (talk) 15:51, 30 April 2008 (UTC)

There were still errors and poor features. I have removed the plot.[6] PrimeHunter (talk) 00:01, 18 August 2008 (UTC)

Identity?

I had a somewhat quick breeze through the article and the talk page archives and I couldn't find this mentioned anywhere (see picture). The sums of the diagonals shown are equivalent to the next number in the sequence minus one. I haven't got around to finding a proof for it yet, though :-(.--Steven Weston (talk) 09:34, 16 July 2008 (UTC)

Actually, it was a very simple matter of proof by induction. Unless anyone is against it, I might put it in the identities section within the next week.--Steven Weston (talk) 12:17, 17 July 2008 (UTC)

It's acceptable in the article if you have a source for it; otherwise it's WP:Original Research. Dicklyon (talk) 14:54, 17 July 2008 (UTC)

I can see how "original research" exists in medicine and other sciences, but when something like this is as basic as 2+2, I see no gain in hiding people from an interesting and verifiable truth, which can be sourced simply and directly from the very axioms of mathematics. A child could verify this logic, whereas I can see how a supposed discovery of some new elementary particle could be classed as original research. I can also see original research in some obscure theorem in the far reaches of group theory. But, if it so pleases the bureaucracy, we could ask the No original research noticeboard. If they decree that it is, I guess someone will have to find someone else's original research that will say exactly the same. Furthermore, Gandalf61's proof without words may actually need some words, or an animation...--Steven Weston (talk) 21:11, 17 July 2008 (UTC)

In this case it's not original research, anyway. It's the second and third formulas in the book "Fibonacci Numbers" by Mircea Martin. —David Eppstein (talk) 22:30, 17 July 2008 (UTC)

Thank you. Glad this is resolved. If no-one puts it up by Monday, I'll do it then. Though I don't have the book, so if someone can source it after I'm done, that'd be appreciated.--Steven Weston (talk) 00:15, 18 July 2008 (UTC)

Here is the book. You'll need to find the page... Dicklyon (talk) 00:28, 18 July 2008 (UTC)

Article title

Shouldn't the title of this article be "Fibonacci numbers" rather than "Fibonacci number"? The term really mainly refers to the sequence rather than to its individual members. Nsk92 (talk) 02:19, 28 August 2008 (UTC)

Wikipedia:Naming conventions#Prefer singular nouns. PrimeHunter (talk) 02:22, 28 August 2008 (UTC)

I think that the part of this convention which is relevant here is where it talks about a small class, such as Arabic numerals, polar coordinates, etc. The term "Fibonacci numbers" usually refers to the entire sequence (and in this sense functions as singular), not to its individual members. The first sentence of the article, where the term is defined, correctly reflects this fact. In fact, mathematically, "Fibonacci sequence" is probably a better term but "Fibonacci numbers" is a more widespread one for historical reasons. This is a different situation from, say, a prime number, since prime numbers are defined by their intrinsic properties, rather than by the order in which they appear in some sequence. It is perfectly fine to say, for example: "17 is a prime number". However, saying "610 is a Fibonacci number", while not incorrect, is fairly unusual. (Certainly more unusual than to say that "5 is an Arabic numeral"). Nsk92 (talk) 02:58, 28 August 2008 (UTC)

F60

is F60 a square number or just close to being one —Preceding unsigned comment added by 124.171.62.10 (talk) 09:32, 18 September 2008 (UTC)

Just close.

F60 = 1548008755920,

1244190^2 = 1548008756100,

1244190^2 − F60 = 180. Gandalf61 (talk) 09:52, 18 September 2008 (UTC)

Fibonnaci Waltz

http://web.tampabay.rr.com/warhawks/FibonacciWaltz.html A clever use of the sequence that should be incorporated into the article.

I don't think it should be added without other sources. This is just one person's invention - it might very well have no significance whatsoever. 165.123.224.192 (talk) 06:30, 3 October 2008 (UTC)

0 first number

This means that 5 is the sixth number of this sequence, not the fifth, and thus the sequence of prime Fibonacci numbers corresponds to 4, 5, 6, 8, 12, 14, 18, 24... not 3, 4, 5, 7, 11, 13, 17, 23.... Georgia guy (talk) 14:42, 8 August 2008 (UTC)

The index of 0 is 0 so F₀=0, F₁=1, ... F₅=5 etc. and the indexes of the prime Fibonacci numbers are indeed 3, 4, 5, 7, 11 ... (sequence A001605 in the OEIS). Gandalf61 (talk) 15:22, 8 August 2008 (UTC)

Tsuris (talk) 06:43, 24 September 2008 (UTC) I have a question about the idea of a "first number", as I was told, the sequence definition is a where every number is the sum of the two preceding numbers, this is usually started off at 1 (not 0 which leads to 0, 0+nothing is 0, 0,0), but as far as I was told, could be any non zero number, .5, .5, 1, 1.5 so forth as an example. —Preceding unsigned comment added by Tsuris (talk • contribs)

It is only called the Fibonacci sequence if it includes two consecutive 1's. As Fibonacci number#Generalizations says, other sequences with other starting values have been studied. PrimeHunter (talk) 10:55, 24 September 2008 (UTC)

Fibonacci fractal

The page 'Fibonacci Fractal' redirects here, but there is no mention of Fibonacci fractal anywhere in the article. Can anyone add something about Fibonacci fractals? - A —Preceding unsigned comment added by 81.101.44.107 (talk) 18:17, 1 November 2008 (UTC)

Maybe the redirect should just be deleted instead. It was originally a one-line stub created by User:Jean-claude perez.[7]. Perez has many times tried to add mention of his own fractal work to Fibonacci numbers. Several other editors have reverted it. Perez also created Fibonacci numbers and Fractals which was deleted at Wikipedia:Articles for deletion/Fibonacci numbers and Fractals. PrimeHunter (talk) 19:39, 1 November 2008 (UTC)

Pythagorean Triples Note

Since I cannot edit the page, maybe somebody could put this into the article if you think it's worth it. It's a little confusing, so bear with me. As is stated in the article, every other number, starting with 5 is the largest in a Pythagorean Triple. There is a formula for finding proving that which is c = m^2 + n^2. (See Pythagorean Triple). m and n are both also in the Fibonacci series consecutively such that their indexes add up to give you the index of the Pythagorean triple. (For example, take 13, which is the 7th number in the series. It's m and n are 2 and 3, which are the 3rd and 4th numbers in the series, and 3 + 4 = 7). I'm not sure if that made sense, but I think that's it's definitely worth trying to fit into the article somehow if it can be worded better. Apmcleod (talk) 02:23, 13 November 2008 (UTC)

regarding external links

can i add my online calculator? i think it's worth it, considering it's WAY faster than other online calculators (especially the one that was here before), and i don't make any money because there are no ads —Preceding unsigned comment added by Kerio00 (talk • contribs) 19:05, 28 January 2009 (UTC)

See WP:EL, in general links don't improve articles content does.TheRingess (talk) 20:25, 28 January 2009 (UTC)

I know, but having at least ONE online calculator is useful, IMHO (especially if someone is actually looking for a calculator, but is too lazy to look up on google or dmoz) Kerio00 (talk) 15:37, 29 January 2009 (UTC)

If somebody needs a huge Fibonacci number then they probably already have a mathematical program to compute them, or can easily find what they need. A calculator with no significant information about Fibonacci numbers beyond the article does not appear useful enough for an external link. PrimeHunter (talk) 16:41, 29 January 2009 (UTC)

There's actually an explanation regarding the algorithm, both in mathematical terms and in Python source code, so that would be also nice for coders. Kerio00 (talk) 17:49, 29 January 2009 (UTC)

Composite Fibonacci numbers

Under "Fibonacci Primes", the article states, "There are arbitrarily long runs of composite numbers and therefore also of composite Fibonacci numbers." I do not know if there exist arbitrarily long runs of composite Fibonacci numbers, but the reason given is insufficient. There exist arbitrarily long runs of non-Fibonacci numbers, so the existence of arbitrarily long runs of numbers having any given property does not imply the existence of arbitrarily long runs of Fibonacci numbers having that property. If this were so, it would mean there are arbitrarily long runs of Fibonacci numbers which are not Fibonacci numbers, which is not the case. Can someone make this assertion more rigorous? Trouserman (talk) 21:35, 7 February 2009 (UTC)

It is already rigorous. If x is composite, so is F_x, as the article already states. n!+2, n!+3, n!+4, ..., n!+n are all composite, so F_n!+2, F_n!+3, F_n!+4, ..., F_n!+n are also all composite. —David Eppstein (talk) 22:23, 7 February 2009 (UTC)

Origins

The rabbit population example is inconsistently presented: first the rule is that every pair has two pairs of offspring and then dies, but then the recursive relation is justified in terms of rabbit fertility. Both schemes (two offspring pairs then death vs. offspring every month from the second month on) give rise to the same sequence, but the presentation should not mix both. Should I fix this? 139.19.84.14 (talk) 17:08, 15 February 2009 (UTC)

recognizing

1st way to recognize it listed in the article makes no sence at all, it is just as good as table look-up or, if no table is available, re-computing all the numbers up to z from scratch. 95.132.178.230 (talk) 14:04, 24 February 2009 (UTC)

No, it is useful. If you want to find a Fibonacci number close to 1000, for example, you do:

{\bigg \lfloor }\log _{\varphi }(1000{\sqrt {5}})+{\frac {1}{2}}{\bigg \rfloor }=16

then you reverse the process:

{\bigg \lfloor }{\frac {\varphi ^{16}}{\sqrt {5}}}+{\frac {1}{2}}{\bigg \rfloor }=987

which not only tells you that 1000 is not a Fibonacci number, but also that 987 is a Fibonacci number. Gandalf61 (talk) 15:43, 24 February 2009 (UTC)

but it actually says F(16) in the article, not that you should "reverse the process"; it is not obvious that such reversal will give you F(16) and not F(16) +1 or -1. 95.132.178.230 (talk) —Preceding undated comment was added on 22:46, 24 February 2009 (UTC).

F(n) is always the closest integer to

{\frac {\varphi ^{n}}{\sqrt {5}}}

. Or, if n is large and you want to stick to integer operations, then F(n) is value of the off-diagonal entries of

{\begin{pmatrix}1&1\\1&0\end{pmatrix}}^{n}

, which you can calculate with an exponentiation by squaring method. So there are definitely methods of calculating F(n) that do not require the calculation of all the preceeding Fibonacci numbers. Gandalf61 (talk) 10:50, 25 February 2009 (UTC)

Removed C program section - explanation

I removed the following new section "Fibonacci series in C" from the article because:

It is unsourced, and so likely to be OR.
It is inconsistent with the convention in the article that F0=0, F1=1.
A recursive algorithm is very inefficient for producing a sequential table of values.

Gandalf61 (talk) 11:26, 9 March 2009 (UTC)

Fibonacci series in C

This is an example of how to print a Fibonacci series in C:

   
#include<stdio.h>   
#include<conio.h>   
int n;   
int fib(int n);   
void main (void)   
{   
clrscr();   
int t;   
printf("\t\tTHE SERIES IS\n");   
for(n=1;n<=25;n++)   
{   
t=fib(n);   
printf("%d , ",t);   
}getch();   
}   
   
int fib(int n)   
{   
if(n==1||n==2)   
return n-1;   
else   
return fib(n-1)+fib(n-2);   
}

MathWorld Fibonacci article should be linked

It's very good, has stuff we don't have, and is as justifiable as present external links. Thanks for considering this suggestion.Rich (talk) 01:03, 12 May 2009 (UTC)

There is a link to the MathWorld article under Notes, reference 15. Gandalf61 (talk) 09:23, 12 May 2009 (UTC)

I know. I say there should be one in the external links, even so. Best wishes, 75.45.125.91 (talk) 15:29, 12 May 2009 (UTC)Rich (talk) 15:31, 12 May 2009 (UTC)

True or False??

True or false: It has been proven that 149 is the largest prime Tribonacci number.

False, there are larger prime Tribonacci numbers, e.g. 19341322569415713958901, 15762679542071167858843489 and 145082467753351661438130501937754420584096000083183992629 (sequence A092836 in the OEIS) Trewal (talk) 12:17, 2 June 2009 (UTC)

somebody. pls translate this to alphabets

1.144.17711_1.5.1_(2)_8.144.987.4181_1.121393.1.377 5.17711.1-5.17711.1_610.1.144_6765.8.377.2.8.233.55.34 2.987.233.8.34_46368.1.10946_4181.8.610.5.1.610.21 34.1.34.1

10946.55.377 1.144.17711_6765.8.2.8.610.1.4181.610.121393.1_ 6765.1.121393.1.610.21.144.1.610_144.1.17711 377.17711.610.21.144.55.610_1.144.17711_10946.8.4181.233.1.233.17711_ 8.377.987 _10946.1.1597.55 89.1.17711.34_5.1.233.1.377_34.1.10946.55, 1.144.17711_6765.1.121393.1.610.21.144.1.610_144.1.17711. 377.1.1.13.144.1.610_1.144.17711_144.1.233.1.17711_ 144.1.10946.1.-144.1.10946.1_1.144.17711 46368.1.10946_144.1.17711_10946.8.4181.233.17711.144.1 -end-

6765.1.1597.1_2.987.233.8.34_10946.8.144.1_5.55.1_ 377.377.21_1597.1.610.5.1.55 —Preceding unsigned comment added by 121.121.67.100 (talk) 08:58, 9 June 2009 (UTC)

It's off-topic, but I enjoyed the puzzle. It seems to be some sort of message in Malay (according to running Google's language detection on my result), in a code that maps letters of the alphabet to Fibonacci numbers (A=1, B=2, C=3, D=5...). If I have the mapping right, it says:

AKU ADA (B) EKOS AYAM DUA-DUA NAK SEMBELIH BOLEH WAT SENDANG HAHA

TIM AKU SEBENASNYA SAYANGKAN KAU MUNGKIN AKU TESLALU EMO TAQI JAUH DALAM HATI, AKU SAYANGKAN KAU MAAFKAN AKU KALAU KATA-KATA AKU WAT KAU TESLUKA -end-

SAQA BOLEH TEKA DIA MMG QANDAI

And now I'm curious what it means. rspεεr (talk) 17:09, 14 June 2009 (UTC)

Fibonicci.co.uk/number-sequences removed discussion.

Hello,

I Got a message that i should discuss the removal of the link.

I don't see why this link does not contribute to the Fibonacci article.

1) The website's name www.fibonicci.co.uk in itself is a referral to the fibonacci sequence 2) Also the logo, of a nautilus shape, is based on the fibonacci sequence 3) The test them self, the medium and the hard one, contains several sequences based that mimic the fibonacci secuence. 4) As far as I looked, the structure of all tests are build up in 8, 13, and 21 questions for the easy, medium and hard tests. Same as fibonacci sequence.

So for me its unclear why this link doesn't belong in the fibonacci article. Especially when compared to the 1st link: On-Line Encyclopedia of Integer Sequences. This has almost nothing to do with the fibonacci sequence. The only relationship being that it is a sequence.

I havent measured the outline of the site, but wouldnt be surprised if somewhere also the golden ratio is applied. At first glance I can understand that maybe the site doesn't belong to the article, but when looked a bit deeper and taken into account all the referals to fibonacci, than in my opinion this is a valid link.

http://www.fibonicci.co.uk/number-sequences

Just my two cents..

Exodian (talk) 19:45, 11 June 2009 (UTC)

See WP:EL. — Miym (talk) 20:36, 11 June 2009 (UTC)

1/89 = Fibonacci Sequence

Interesting fact: http://www.geom.uiuc.edu/~rminer/1over89
Maybe this could be added to the article? --41.204.111.27 (talk) 00:27, 12 June 2009 (UTC)

It is already mention in the section Power series, but I have added your link as a source. Gandalf61 (talk) 14:07, 12 June 2009 (UTC)

Indeed, many editors get confused about these two articles; the remarkable number 1/89 is already in Fibonacci_number#Power_series, with a ref, but done up in a way that make the relationship described above very hard to see, so that might be a good thing to improve on. Dicklyon (talk) 03:37, 25 September 2009 (UTC)

fibonacci when did he invent it and why

who what when where and why i need these answers —Preceding unsigned comment added by 94.0.158.49 (talk) 15:14, 29 September 2009 (UTC)

Who, what and when are all in the first few lines of this article. For where and why, see our article on Liber Abaci. Gandalf61 (talk) 15:33, 29 September 2009 (UTC)

Simple Cell division rate as explanation for Fibonacci numbers and for golden mean in nature.

[8]

On the first topic thread there is the proposal, and linked simple diagram. Then follows open discussion, which at some point turns into computed tests to disprove or prove the explanation. Thus far no fault seems to have been found in it. Seems deceptively simple...

In a nutshell: an organism, like cell (or even self replicating bubble ?) is born. Then it exists and grows. Then it divides, gives birth to another organism. While the new "child" organism is still gathering energy and growing, the original orgnism splits again. Then there is three. Next the elder child organism and parent organism split while youngest one still grows, and there is 5 of them, and it goes on and on and seems to give the fibonacci number each time, as well as the golden proportion in shapes and volumes, even if some of the organisms get wrecked at some generations, or starting amount of organisms is varied.

Direct link to the cell division diagram only, without the discussions, commentary and testing: [9]

My sources have been... I just started to calculate it as visual organic division, without numbers. I think I was trying to understand development of organic spatial forms. No other sources, except that I did not seem to find quite this simple explanation that gives reproducible results, in wikipedia or elsewhere in the Net. At least not for now.

It is so very simple that it may be wondered why I post it... and the answer is, because I have not yet seen it elsewhere, only more complex models with more parameters, and things like rabbits and bee communities ratios of young queens.

--MaxTperson (talk) 14:49, 12 January 2010 (UTC)

13 Popular Culture [[10]]

No mention here of the popular culture that exists in the Stock Trading business, Economics, Mysticism and other parts of the culture that are continually on the look out for Fibonacci Numbers in any series - it is because it is a "Natural" Series. Just look at the external links to see just how wide spread and deeply influential in popular culture this series is.

Good point! Also Bartok and other composers! 75.48.16.182 (talk) 20:09, 2 October 2010 (UTC)

F(x) for fabonci

f(x) = exp(0.481219*x - 0.8049) + 0.00114292*x^2 - -0.355997^x - 0.0112628*x

or a bit more complex

f(x) = exp(0.481219*x - 0.804903) + 0.00160452*x^2 + 0.0126129*log(x) - -0.364351^x - 0.00160452*x*log(x) - 0.014535*x - 0.00226154

(amazing results pretty close isn't it ??) —Preceding unsigned comment added by 82.217.115.160 (talk) 17:12, 3 April 2010 (UTC)

Weird, I did not post this... starting from " F(x) for fabonci " is not my text or opinions at all. Why is it on my signature ? —Preceding unsigned comment added by MaxTperson (talk • contribs) 22:20, 11 October 2010 (UTC)

Probably because whoever did post it put it between your post and its signature. I've moved your sig up; please check whether it is now in the right place. —David Eppstein (talk) 22:25, 11 October 2010 (UTC)

Incorrect Equivalence

See my comments on [11]. --Andreas Rejbrand (talk) 22:47, 12 May 2010 (UTC)

Easy way but take time

The pattern is like this
0 + 1 = 1
1 + 1 = 2
2 + 1 = 3
3 + 2 = 5
5 + 3 = 8
8 + 5 = 13
13 + 8 = 21
21 + 13 = 34
34 + 21 = 55

How many year you finish to the numbers 100? --Samit Boonyaruk (talk) 14:29, 24 August 2010 (UTC)

It gets too large for a regular calculator (3.5422484817926E+20), but with pen and paper it may be nice for a rainy day.--Patrick (talk) 14:50, 24 August 2010 (UTC)

A matter of style

In the article we have: "the (n + 2)nd Fibonacci number". Why "nd"? Is it because it's preceded by a "2"? To stop people saying "n plus tooth"? I think "th" is better.

But the article isn't consistent: lower down we have: "we consider the (n + 1)th summand". To be consistent it would say: "(n + 1)st". I think "th" is better.

Show of hands for an edit? Change "(n + 2)nd" to "(n + 2)th"? R L Lacchin (Gloucester, UK) (talk) 14:05, 28 October 2010 (UTC)

I think the two-letter suffix is "stretching" the usage too far. I'd say that a more-elaborate construction is better. (Fwiw, I was employed as an editor for a couple of years, and had very few copy-edit changes to what I typed.) Nikevich (talk) 06:07, 29 November 2010 (UTC)

Contradicting Connotations?

In the "In nature" section, there seem to be some balancing problems in the first paragraph:

"Fibonacci sequences appear in biological settings, [various examples of the sequence in nature]. In addition, numerous poorly substantiated claims of Fibonacci numbers or golden sections in nature are found in popular sources..."

The use of "In addition" and "poorly" is slightly awkward and contradictory. I'd recommend something like "There are numerous claims of Fibonacci numbers or golden sections in nature, although many are poorly substantiated." Thoughts? 174.91.190.188 (talk) 03:25, 13 December 2010 (UTC)

Okay with me. —Mark Dominus (talk) 04:19, 13 December 2010 (UTC)

Systematic bias in revert by David Eppstein

This is related to the revert by David Eppstein and the prejudiced remarks with which a cited fact was reverted, along with other edits.

The name Gopala-Hemachandra number is gaining currency as a general form of the Fibonacci number. Two mathematics papers, one from the J of Number theory, were cited to show how this term is used.

Whether the term is "a trivial extension" is not an issue on Wikipedia. The fact is that it is being used, and we need to report it here.

If you had argued about the citations, said that they are non-authoritative, showed other evidence that these are not coherent or otherwise, I would have understood this. On wikipedia it doesn't suffice to say that the editor has an Indian-sounding name and hence this is Nationalistic edit.

In fact, there is stronger evidence for Wikipedia:Systemic bias (eurocentric bias) in this revert than any factual stance.

The revert also had other edits other than this issue, such as the incorrect "disputed" characterization. Please re-edit the text, if needed, do not revert.

At the very least, such issues should be discussed here before reverting edits.

I am reverting the revert and will expect a civilized response on this page rather than a revert war. mukerjee (talk) 06:41, 26 March 2011 (UTC)

The phrase "Gopala-Hemachandra number" has exactly zero hits in Google scholar. And if it had two hits, that would still be a tiny number compared to the vast literature on Fibonacci numbers. As for the disputed part, please read the paragraph in the "Origins" section, which clearly outlines the dispute among scholars about the timing of the discovery in India of the Fibonacci numbers, and clearly contradicts your poorly-sourced edits in the lead section.. —David Eppstein (talk) 06:49, 26 March 2011 (UTC)

I get four hits, including the Basu-Prasad paper, the Thomas paper, and two more. Is gscholar geography-biased? mukerjee (talk) 07:16, 26 March 2011 (UTC)

I reviewed the first revert carefully, as it seemed that it might be better to include this material in the article. The more I looked at it though, the more it became clear that a much more complete and well referenced treatment of the contributions of these two guys was already in the article. There's no need to put this obscure and controversial bit into the lead as well. Dicklyon (talk) 06:54, 26 March 2011 (UTC)

David:

You make the point

nationalist claims of priority that are sourceable but also disputed in the literature.

Pls cite a source that "disputes" the Indian prior claim. I can't find any and would be happy to have some. The three articles cited in the article all dispute whether the Indian work dates from 400BC or 700BC, no one argues against any priority.

But this is an aside. The main point is that there are references using this name.

I am attaching some segments of the paper. It is available through elsevier at http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WKD-4YT7KMP-1-1&_cdi=6904&_user=489944&_pii=S0022314X10000533&_origin=gateway&_coverDate=09%2F30%2F2010&_sk=998699990&view=c&wchp=dGLzVlz-zSkzk&md5=f2a86d5e45ce40cf5875f53017c9befb&ie=/sdarticle.pdf

Here sre some extracts from the Basu-Prasad paper:

Journal of Number Theory 130 (2010) 1925–1931

Long range variations on the Fibonacci universal code

Manjusri Basu, Bandhu Prasad Received 15 September 2009 Revised 6 January 2010 Available online 8 April 2010 Communicated by David Goss

Abstract

Fibonacci coding is based on Fibonacci numbers and was deﬁned by Apostolico nd Fraenkel (1987) [1]. Fibonacci numbers are generated by the recurrence relation Fi = Fi−1 + Fi−2 ∀i >= 2 with initial terms F0 = 1, F1 = 1. Variations on the Fibonacci coding are used in source coding as well as in cryptography. this paper, we have extended the table given by Thomas [8] We have found that there is no Gopala–Hemachandra code for a particular positive integer n and for a particular value of a ∈ Z . We conclude that for n = 1, 2, 3, 4, Gopala–Hemachandra code exists for a = −2, −3, . . . , −20. Also, for 1 � n � 100, there is at most m consecutive not available (N/A) Gopala–Hemachandra code in GH−(4+m) column where 1 � m � 16. And, for 1 � n � 100, as m increases the availability of Gopala–Hemachandra code decreases in GH−(4+m) column where 1 � m � 16.

... And section 2 has:

2. Gopala–Hemachandra (GH) sequence and codes

A variation to the Fibonacci sequence is the more general GH sequence [6] {a, b, a + b, a + 2b, 2a + 3b, 3a + 5b, . . .} for any pair a, b which for the case a = 1, b = 2 represents the Fibonacci numbers [4,5,7] gives historical details of these sequences.

The paper is available from Elsevier at http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WKD-4YT7KMP-1-1&_cdi=6904 (or at most univ libraries)mukerjee (talk) 07:16, 26 March 2011 (UTC)

Origins section

I hate edit wars.

re-reading the whole article, I find that the origin section has the sentence, which is perhaps part of what Dicklyon is referring to:

However, according to Knuth, neither of these works concern the more specific problem of counting patterns with a fixed total duration, for which the answer is the Fibonacci numbers, and for which Knuth cites instead the anonymous Prakrta Paingala (c. 1320 AD), which uses Fibonacci coding to solve the problem.

But what Knuth says in the relevant part is:

... consider the set of all sequences of L and S that have exactly m beats. ... there are eactly Fm+1 of them. For example the 21 sequences when m=7 are: [List]

In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8

Moreover, the anonymous author of prAkr.ta paingala (c.1320) discovered elegant algorithms for ranking and unranking w.r.t. m-beat rhythms. p. 50

Thus the statement "neither of these works concern the more specific problem of counting patterns with a fixed total duration, for which the answer is the Fibonacci numbers" appears to be a deliberate misleading the reader. In fact, Knuth says unequivocally in v.1 p.100:

Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns... both Gopala (before 1135AD) and Hemachandra (c.1150) mentioned the numbers 1,2,3,5,8,13,21 ... explicitly. [See P. Singh Historia Math 12 (1985) 229-244] AOP v.1 3d ed. p. 100

I am not sure who inserted this sentence, but if there is evidence of prejudice it is in this sentence, which is also used to make an incorrect claim about what was disputed in the lead.

Would anyone object if I edited the origins section, or is the above text justified? mukerjee (talk) 09:01, 26 March 2011 (UTC)

I don't have immediate access to a recent edition of Knuth Vol.I which is what you seem to be quoting; the part about Gopala and Hemachandra certainly isn't in the part of Knuth that is cited in the article. Maybe someone else who does have a recent edition and can check what Mukerjee says would like to weigh in? —David Eppstein (talk) 23:11, 26 March 2011 (UTC)

there is no problem accessing AOP - i have also provided the google books links.

Even if others haven't accessed it yet, it follows wikipedia verifiability norms so I am going ahead and editing in the changes.

Pls note that wikipedia policy suggests: "Please boldly add information to Wikipedia, either by creating new articles or adding to existing articles, and exercise particular caution when considering removing information."

So pls take added care in removing edits that contain verifiable facts.

I will next be re-inserting some of the original text in the lead. mukerjee (talk)

Sunflower image

I removed the sunflower image since when I tried to verify the spiral count I kept getting different results. I think the particular sunflower in the image had irregularities that made the spirals too unstable to make the counts meaningful. The other sunflower images I found weren't much better but I did find a chamomile image where I could draw in the spirals. I also removed the seashell image, the connection to the Fibonacci number has been claimed by some authors but my understanding is that this has been debunked.--RDBury (talk) 02:05, 28 April 2011 (UTC)

First number

As the article notes, the Fibonacci sequence is sometimes said to start with 0, sometimes with 1. Strictly speaking, if you are calling this sequence the "Fibonacci" sequence, you should begin with 1 because that is how Fibonacci started the sequence. Might I suggest you start with 1 and note that "some sources start with 0". In addition to historical accuracy, I suspect starting with 1 is far more popular in the literature, despite the OEIS entry. Or perhaps it would require too much rewriting of the article to make it worth the trouble to be historically accurate? --seberle (talk) 06:06, 26 February 2011 (UTC)

I admit this is probably minor, but it still bugs me that starting with zero is not, very strictly speaking, the "Fibonacci" sequence, nor is it the sequence that most readers will come across in other readings. At the very least, could we replace "Some sources omit the initial 0" in the introduction with "Fibonacci originally started his sequence with two 1s instead of 0 and 1, and many still define the sequence that way instead"? The way it's written now makes it look as if starting with zero were the definitive way and starting with two 1s is an infrequent variation. --seberle (talk) 17:12, 27 February 2011 (UTC)

I don't think Fibonacci called it "the Fibonacci sequence". It was named later, and named objects often deviate from the work of the person they were named after. Including 0 is mathematically important for many of the formulas and should be "our" definition, but we can reformulate the description of the alternative definition. It currently says: "Some sources omit the initial 0, instead beginning the sequence with two 1s". That can indeed give a misleading impression of the history and prevalence. I suggest: "Many sources do not include an initial 0, instead beginning the sequence with two 1s". PrimeHunter (talk) 17:54, 27 February 2011 (UTC)

There are two slightly different issues here:

(A) Should the enumeration of the sequence start with

F_{0}=0\quad {\text{and}}\quad F_{1}=1\,,

or with

F_{0}=1\quad {\text{and}}\quad F_{1}=1\,.

(B) Should the quoted sequence of values start 0,1,1,... or start 1,1,2,...

On (A), it is clear that the enumeration must start with

F_{0}=0

; if we change this, many of the quoted properties of the sequence do not hold (for example, F_kn is no longer divisibel by F_n). On (B), I have no preference. On either point, I think that the argument "this is not how Fibonacci defined the sequence" does not carry much weight - many mathematical objects and terms have modern definitions that differ widely from those given by their originators. Fibonacci's original formulation may be of historical interest but it should not dictate how we write our article. Gandalf61 (talk) 10:39, 28 February 2011 (UTC)

I agree. Clearly the best way to define the Fibonacci sequence is as it is done in Wikipedia. However, we are still dealing with two problems. (This is all rather minor really, a question of wording in the introduction.) (1) The sequence was historically defined as beginning 1, 1, ... This goes beyond how Fibonacci defined the sequence (and yes of course he didn't name it after himself -- he didn't name it all as far as I know) because (2) many sources (and I strongly suspect the majority of popular sources) still begin with 1, 1, ... today; this is not some archaic historical distinction. Therefore the remark "Some sources omit the initial 0", as if starting with 1 was an afterthought, is rather misleading. Though not intentionally stated, the wording gives the impression that those who start with 1 are doing something wrong. PrimeHunter's proposal is an improvement, though it still implies something is being left out, when actually 0 is being added to the original sequence for the reasons Gandalf61 points out. And it would still be nice to include that this is how the sequence was historically defined (and not vice versa). It really shouldn't be hard to come up with a better wording for this one sentence. --seberle (talk) 16:30, 28 February 2011 (UTC)

The separation into (A) and (B) is good. I agree with the first choice in (A). This should not be too controversial though it isn't done by everyone.

Here's what I remember from what I've seen in math and science, which may be correct: "the Fibonacci numbers" usually start with 1, 1 and then usually they're called F₁, F₂. The math people would generally be happy to include F₀ = 0. Some would like to say F₀ = 1, F₁ = 1. Some experts would say it depends on what suits the context.

I do think most people who know the sequence would start it 1,1,....

Here is an argument for not including 0 in "the" (basic) sequence. The number f_n of sequences of 1's and 2's of total length n starts with f₀ = 1, not 0. The value 0 doesn't have any place in this count. Many other things counted by Fibonacci numbers behave the same way. I don't doubt there are counterexamples; I'm just mentioning that this is one justification, in addition to the original Fibonacci problem, for preferring to start at 1, or at least not preferring to start at 0. In my experience the single best reason for including 0 in the sequence is the attractiveness of working backwards via the recurrence relation.

My suggestion would be to revise the first few lines to say the sequence is 1,1,2,3,..., then say some prefer 0,1,1,2,3,..., but mathematically the conventional recurrence may start with initial conditions either F₀ = 0, F₁ = 1 or F₁ = 1, F₂ = 1; and furthermore some people displace the sequence and say F₀ = 1, F₁ = 1. This would be true to real life, which would probably be more useful to ordinary users than a single rigid definition. Zaslav (talk) 03:06, 26 March 2011 (UTC)

I believe that the history of how the Fibonacci series came about informs this discussion. That history is already in the article under the heading Origins. But Origins should start with the Fibonacci story if we are to retain the word Fibonacci in the title of the entry. The Indian mathematics which developed the same series at an earlier time should appear but not before the Fibonacci history is told. To do this in no way diminishes the Indian achievement - it just places it in the context of an article on the Fibonacci series. Fibonacci himself developed the series to do a job. That job was to calculate how many rabbits you would have after a year if you took as your data the number of kittens rabbits usually have, and how often they are fertile and assuming that on average they have equal numbers of male and female offspring. The problem thus starts with a pair of rabbits. So the series as conceived by Fibonacci starts with 1. (one pair of rabbits) Some might think that 0 should be the first number so that the series can start using the rule F_n = F_n-1 + F_n-2. If we use this rule and start as Fibonacci started then we start with 1 and add the number which came before it. No number came before it so add zero. 1+0=1 the second number. And so on. But if we start with zero then the series gets nowhere if we wish to find the second number by the rule because if we start with 0 then the next number must be 0+0=0 and the series gets nowhere. Even the Indian series starts with a whole number greater than zero. So starting with zero is really not only historically incorrect but conceptually incorrect and it is thus misleading as well. This leads me to the idea that the article might be renamed The Fibonacci Series or Fibonacci Numbers. In order to recognise the common but historically and conceptually incorrect way of developing the series a footnote or a comment might be made that "some people after Fibonacci used to start and some still start the series with zero" and then the article could go on to comment that this is historically and conceptually correct and misleading. Robarc (talk) 17:22, 12 June 2011 (UTC)

In view of the discussion above I am changing the introduction to state that the series begins 1, 1 or 0, 1, depending on personal preference. Zaslav (talk) 05:52, 28 June 2011 (UTC)

It seemed best to allow that the sequence may begin 1,1 or 0,1, and the numbering of the sequence may begin F₀,F₁ in either case or F₁,F₁ in the former case. There is no general agreement about which to use. I stated that in this article 0,1 is used. Zaslav (talk) 06:15, 28 June 2011 (UTC)

Relation to Golden Ratio Information

I am not a mathematician, but the following sentence in the information beneath the image in the Golden Ratio section does not properly describe the equations in the image: "The length of the side of one square divided by that of the next smaller square is the golden ratio." Could someone that knows better fix either the image or the text?--Bainst (talk) 19:00, 1 May 2011 (UTC)

The quoted caption at Fibonacci number#Relation to the golden ratio looks right to me. 1 / (1/φ) = φ (this holds for any non-zero number and not just φ). And more generally, (1/φⁿ) / (1/φⁿ⁺¹) = φ (also holds for all non-zero numbers). In addition, φ has the special property 1/φ = φ−1. PrimeHunter (talk) 01:15, 2 May 2011 (UTC)

"Gopala-Hemachandra Numbers"

My remarks on Talk:Gopala–Hemachandra number seem to be relevant, so I am copying them here. I said:

There does not seem to be any source for the main claim of this article, namely that "A Gopala–Hemachandra number is a term in a sequence of the form …." The sources cited in the article do not state this. I cannot find any indication that anyone actually uses the term "Gopala–Hemachandra number" in this way, or indeed for anything else.

I am not disputing that Gopala and Hemchandra dicussed the Fibonacci series before Fibonacci did; I agree that that is well-established. My only complaint is with the claim that the term "Gopala–Hemachandra number" is a recognized term.

Two of the cited sources refer to "Gopala–Hemachandra codes". I don't think these two sources are enough to establish that the term is widely used. One of the two sources cited for this is self-published, and does not meet Wikipedia's standards for reliable sources. —Mark Dominus (talk) 15:13, 28 March 2011 (UTC)

The J.H. Thomas source does use "Gopala-Hemachandra sequence" to refer to a general sequence that obeys the recursion $a n$ = $a n -2$ + $a n -1$ , but it is self-published. I have not seen the Basu-Prasad paper yet; I have written to Professor Basu asking for a copy, but she has not yet replied. But even if it does use "Gopala-Hemachandra sequence" or "Gopala-Hemachandra number", it's only a single paper; the term is clearly not in widespread use, and I think it is inappropriate to add it to this article as if it were a common phrase. —Mark Dominus (talk) 16:00, 29 March 2011 (UTC)

Just about everyone in the world says "Fibonacci number". I hate it when people claim we can't use the standard terminology for some newly discovered historical reason. The result, when the new name is used, is that finding literature becomes substantially and unnecessarily more difficult. Attempts at changing names happen regularly, usually when a different European mathematician is claimed to precede the one whose name is used. Often, the justification for the claim is rather unclear, and usually it does not change existing terminology. Thank goodness. Changing very well-established terminology ought to be done only for very good reasons. Zaslav (talk) 06:26, 28 June 2011 (UTC)

Anyway the standard here is to name things what they are named, not what they should be named. See also Stigler's law of eponymy. —David Eppstein (talk) 06:43, 28 June 2011 (UTC)

How a bit of information on the complex plane?

If you use the equations on this page for i, what is the i^th Fibonacci number? Robo37 (talk) 09:43, 6 July 2011 (UTC)

I think it's 0.221247712 + 0.299699204i. Do you think this should be included in the artical?

Can you show the details of your calculation ? Gandalf61 (talk) 13:09, 8 July 2011 (UTC)

The details of my calculation?... I entered (((1+sqrt 5)/2)^i-(-(1+sqrt 5)/2))^i)/sqrt 5 into Google calculator... I am no mathematician, sorry, but maybe if you can look into how Google calculator got to this you might be able to change that into some king of closed solution involving e and/or pi. Robo37 (talk) 13:38, 11 July 2011 (UTC)

Oops, wrong formular. The i^th Fibonacci number is 0.379294534 + 0.215939518 i, which I got by ((1+sqrt 5)/2)^i-((-1)^i/((1+sqrt 5)/2)^i)))/sqrt 5.

Fibonacci root?

What is x when F (x) = n? Robo37 (talk) 09:57, 14 July 2011 (UTC)

See Fibonacci number#Recognizing Fibonacci numbers. PrimeHunter (talk) 11:45, 14 July 2011 (UTC)

Thanks, is it at all expressable without using uncommon logs? Robo37 (talk) 17:58, 15 July 2011 (UTC)

You can always change a logarithm base to any other with the formula at Logarithm#Change of base. PrimeHunter (talk) 02:08, 16 July 2011 (UTC)

Wait, how come when I enter log ((21*sqrt 5)+0.5)/log ((1+sqrt 5)/2) into Google calculator I get 8.02106857? Robo37 (talk) 07:42, 16 July 2011 (UTC)

Because that is the (approximate) value of that expression. Note that to find the Fibonacci index you have to take the floor (nearest integer less than or equal to) this value, which gives you 8. Gandalf61 (talk) 14:56, 16 July 2011 (UTC)

The difference to the real value converges to 0. In PARI/GP for powers of two:

? for(n=1,8,print(2^n": "log((fibonacci(2^n)*sqrt(5))+0.5)/log((1+sqrt(5))/2)))
2: 2.09163988209225188766928942555474042584550500130492803616758
4: 4.10467845877076686103432829279967108337857360119089528434855
8: 8.02106857224470714263573083533459921808575680298676770865263
16: 16.0004703147334956159652160508455984425808456893171492670951
32: 32.0000002133187473657459828226685085556081114874256292899634
64: 64.0000000000000437950208463788551755832830904781992970335194
128: 128.000000000000000000000000001845932267159675589494339813885
256: 256.000000000000000000000000000000000000000000000000000003279

PrimeHunter (talk) 16:25, 16 July 2011 (UTC)

Oh, right, thanks. What is the (closed) exact formular, then? Robo37 (talk) 17:08, 16 July 2011 (UTC)

Well, if you want an expression that does not involve the floor function then it is

n=\log _{\varphi }{\bigg (}{\frac {{\sqrt {5F_{n}^{2}\pm 4}}+{\sqrt {5}}F_{n}}{2}}{\bigg )}

where you choose the sign (plus or minus) that makes

{\sqrt {5F_{n}^{2}\pm 4}}

an integer. Gandalf61 (talk) 19:47, 16 July 2011 (UTC)

Thanks, but that get's to (log (((5*(((1+sqrt 5)/2)^21-((-1)^21/((1+sqrt 5)/2)^21)))/sqrt 5)))+4)+((sqrt 5)*((1+sqrt 5)/2)^21-((-1)^21/((1+sqrt 5)/2)^21)))/sqrt 5)))/2))/(log ((1+sqrt 5)/2)) and when I enter it into Goodle calculater I get 58567.8366. With an equation as massive as that I'm bound to make an error somewhere along the line but there must be a shorter expression than that I can use without using the Fibonacci function itself. Robo37 (talk) 20:17, 16 July 2011 (UTC)

F₈ = 21, so

\log _{\varphi }{\bigg (}{\frac {{\sqrt {5\times 21^{2}+4}}+21{\sqrt {5}}}{2}}{\bigg )}=\log _{\varphi }{\bigg (}{\frac {{\sqrt {2209}}+21{\sqrt {5}}}{2}}{\bigg )}=\log _{\varphi }{\bigg (}{\frac {47+21{\sqrt {5}}}{2}}{\bigg )}=8

Without using the floor function, that's as simple as it gets. Gandalf61 (talk) 21:33, 16 July 2011 (UTC)

Here is a valid ASCII expression to compute x when F (x) = n:

x = log((sqrt(5*n^2+4)+sqrt(5)*n)/2)/log(((1+sqrt(5))/2))

Replace +4 by -4 if required to get an integer (this happens when x is odd). PrimeHunter (talk) 23:02, 16 July 2011 (UTC)

Thanks for that, we got there eventually. So, as I don't see

\log _{\varphi }{\bigg (}{\frac {{\sqrt {5\times n^{2}+4}}+n{\sqrt {5}}}{2}}{\bigg )}=x

and

\log _{\varphi }{\bigg (}{\frac {{\sqrt {5\times n^{2}-4}}+n{\sqrt {5}}}{2}}{\bigg )}=x

anywhere on the artical, maybe it should be included somewhere? Anyway, thanks again, I appreciate the help. Robo37 (talk) 09:06, 17 July 2011 (UTC)

Sorry about this, but is there a universal function that for that that isn't dependant on whether x is odd or even? Robo37 (talk) 22:38, 29 July 2011 (UTC)

On entering decimals into the Fibonacci function..

We have F(x)=((1+sqrt 5)/2)^(x)-((-1)^(x)/((1+sqrt 5)/2)^(x))))/sqrt 5, which indeed satifies any integar as:

((1+sqrt 5)/2)^(0)-((-1)^(0)/((1+sqrt 5)/2)^(0))))/sqrt 5=0
((1+sqrt 5)/2)^(1)-((-1)^(1)/((1+sqrt 5)/2)^(1))))/sqrt 5=1
((1+sqrt 5)/2)^(2)-((-1)^(2)/((1+sqrt 5)/2)^(2))))/sqrt 5=1
((1+sqrt 5)/2)^(3)-((-1)^(3)/((1+sqrt 5)/2)^(3))))/sqrt 5=2
((1+sqrt 5)/2)^(4)-((-1)^(4)/((1+sqrt 5)/2)^(4))))/sqrt 5=3
((1+sqrt 5)/2)^(5)-((-1)^(5)/((1+sqrt 5)/2)^(5))))/sqrt 5=5
((1+sqrt 5)/2)^(6)-((-1)^(6)/((1+sqrt 5)/2)^(6))))/sqrt 5=8...

...and so on, so it thereby would make sense to use the same forumular for decimals.

Using Google calculator, we have:

x	((1+sqrt 5)/2)^(1/x)-((-1)^(1/x)/((1+sqrt 5)/2)^(1/x))))/sqrt 5
1	1
2	0.568864481 - 0.351577584 i
3	0.905958432
4	0.224000793 - 0.280383911 i
5	0.898572747
6	0.127109339 - 0.206373405 i
7	0.896541471
8	0.0858883613 - 0.161150332 i
9	0.895706001
10	0.063917354 - 0.131703889 i
11	0.895283186
12	0.0505164733 - 0.111197646 i
13	0.895040036
14	0.0415832749 - 0.0961519693 i
15	0.894887492
16	0.0352431027 - 0.084662092 i
17	0.89478555
18	0.0305294065 - 0.0756092279 i
19	0.894714073
20	0.0268975698 - 0.0682964386 i
21	0.894662029
22	0.0240191066 - 0.0622681185 i
23	0.894622962
24	0.0216849331 - 0.0572143342 i
25	0.89459289

So we have when x is odd, the number is real and converges to 2/sqrt 5. This is rather interesting as it shows that i is not just the solution to the square root of -1, but also the solution to non-integar Fibonacci numbers, which keeps me wondering what the ith Fibonacci root is? I tried using the obove Fibonacci root function wich leaves 2 + 3.2642513 i when you assume i to be odd or 1 + 3.2642513 i when you assume i to be even, with 3.2642513 aparently being what the rank-1 Grothendieck constant is at most in the tripartite graph G (whatever that means), but when you enter either of those numbers into ((1+sqrt 5)/2)^(1/x)-((-1)^(1/x)/((1+sqrt 5)/2)^(1/x))))/sqrt 5 you do not get i, which suggests the i is neither odd nor even.

Any help will be greatly appreciated. Thanks. Robo37 (talk) 11:15, 31 July 2011 (UTC)

That should be ((1+sqrt 5)/2)^(1/x)-((-1)^(1/x)/(( −1 +sqrt 5)/2)^(1/x))))/sqrt 5. Your biggest problem when trying to extend the Binet formula to non-integer exponents is that when x is not an integer, (-1)^x is not well defined. For example, in the second line of your table, you have used the value i for (-1)^(1/2) rather than -i - this is an arbitrary choice. It is equally valid to say that FR(1/2) could be 0.568864481 + 0.351577584 i. Gandalf61 (talk) 08:03, 9 August 2011 (UTC)

Thanks for the help, but there must be some kind of value of x when ((1+sqrt 5)/2)^(x)-((-1)^(x)/((1+sqrt 5)/2)^(x))))/sqrt 5=i, right?... is there any way to compute this? Robo37 (talk) 10:51, 11 August 2011 (UTC)

Your function seems to get me to -18.1040617 + 24.5918696 i instead of i for some reason. Robo37 (talk) 11:02, 11 August 2011 (UTC)

Example of Implementation in Programming

Seeing as calculation of the Fibonacci Number is a fundamental problem in recursive computer programming, I would find it only appropriate that this page contain at least one example implementation. I feel that many people looking into the Fibonacci Sequence may have some computer background and a quick topic containing some basic code to calculate it, ideally in a functional programming language, would be of great use to them. I would be more than happy to write this up and add it, but as somewhat of a newcomer to the contribution side of Wikipedia I felt it appropriate that I gauge people's feelings on this matter first. Thoughts? — Preceding unsigned comment added by Swat510 (talk • contribs) 07:00, 6 August 2011 (UTC)

Thanks for coming here first. Fibonacci number#External links has the box to the right with a link to a page which was originally a Wikipedia article at Fibonacci number program, but it was transwikied to Wikibooks at Wikipedia:Articles for deletion/Fibonacci number program (2 nomination). There are still several examples at Recursion (computer science)#Fibonacci where it seems more appropriate for the purpose you mention. I have added a link to Fibonacci number#See also. This seems sufficient for this article which is not suppsed to be about programming. PrimeHunter (talk) 14:02, 6 August 2011 (UTC)

Ah, nifty. Didn't see that before. I agree the Recursion topic makes more sense.

^ Posamentier, Alfred (2007). The (Fabulous) FIBONACCI Numbers. Prometheus Books. p. 300. ISBN 978-1-59102-475-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
^ Posamentier, Alfred (2007). The (Fabulous) FIBONACCI Numbers. Prometheus Books. p. 305. ISBN 978-1-59102-475-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[1] Posamentier, Alfred (2007). The (Fabulous) FIBONACCI Numbers. Prometheus Books. p. 300. ISBN 978-1-59102-475-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[2] Posamentier, Alfred (2007). The (Fabulous) FIBONACCI Numbers. Prometheus Books. p. 305. ISBN 978-1-59102-475-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[1]

[2]