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April 29

Sequences of numbers in Pascal's tetrahedron

Pascal's triangle has some interesting sequences, including the all 1's, the powers of 2 (we add all the numbers on the same row to get a power of 2,) the counting numbers, the triangular numbers, and the tetrahedral numbers. What about Pascal's tetrahedron?? I know that the sum of all the numbers on any layer of Pascal's tetrahedron is a power of 3. But what other sequences are related to Pascal's tetrahedron?? Georgia guy (talk) 15:17, 29 April 2023 (UTC)[reply]

OEIS sequence A006480 is found on the central spike. --Lambiam 17:20, 29 April 2023 (UTC)[reply]

You can get the Fibonacci sequence by summing along "diagonals" of Pascal's triangle; see the diagram at the top of the Applications section. Similarly, you can get more complex sequences by summing over diagonal slices of the tetrahedron. For example Pell numbers, Jacobsthal numbers and Tribonacci numbers. --RDBury (talk) 04:37, 30 April 2023 (UTC)[reply]

PS. By summing along diagonal lines you get another triangle, what OEIS calls the "Skew Fibonacci-Pascal triangle", see OEIS: A037027. You can then get some new sequences be reading off individual rows and columns; in one direction you get Convolved Fibonacci sequences. The main diagonal is OEIS: A038112. Basically it's a three dimensional array and there are a lot of ways to slice and dice, and there's a good chance that the results will be in the OEIS. --RDBury (talk) 05:16, 30 April 2023 (UTC)[reply]

PPS. You can get Delannoy numbers by summing along a different set of lines. It's not mentioned in the article, but the row sums of this triangle are the Pell numbers. --RDBury (talk) 05:28, 30 April 2023 (UTC)[reply]

~~The Delannoy numbers also occur as entries in Pascal's pyramid by themselves.~~ --Lambiam 17:41, 30 April 2023 (UTC)[reply]

I'm not seeing that, other than trivial occurrences at n=comb(0, 1, n-1). For example how would 41 and 61 appear? RDBury (talk) 19:53, 30 April 2023 (UTC)[reply]

Sorry, I was mistaken. --Lambiam 13:14, 1 May 2023 (UTC)[reply]

(Link for convenience: Pascal's pyramid and Pascal's simplex. catslash (talk) 13:04, 3 May 2023 (UTC))[reply]

Can you please add any notable ones to the article Pascal's pyramid? Thanks, cmɢʟee⎆τaʟκ 02:11, 4 May 2023 (UTC)[reply]