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December 21

Two mathematical terms sought

What are the accepted terms for the following two types of numbers?

The sequence 1, 3, 6, 10, 15, 21 . . .
Numbers like 456, 2,345, 12,345, 45,678, 345,678, 987, 3,210, 65,432, 765,432, etc. Khemehekis (talk) — Preceding undated comment added 10:46, 21 December 2014 (UTC)[reply]

The first lot are the Triangular numbers. I don't know a special term for numbers where the digits are in sequence. Dmcq (talk) 11:39, 21 December 2014 (UTC)[reply]

Thanks for the term "triangular numbers". As for the latter group, I call them "stairstep numbers" in this thread. I want to know the "real" term, though. Khemehekis (talk) 12:15, 21 December 2014 (UTC)[reply]

OEIS:A138141 ("Numbers with digits in ascending order that differ exactly by 1") and OEIS:A138142 ("Nonnegative numbers with digits in descending order that differ exactly by 1") give no names. PrimeHunter (talk) 12:40, 21 December 2014 (UTC)[reply]

OK, then. I guess I'll just continue to use "stairstep number" (a translation of the Kankonian term bayet na tatz*emad). Much as I use the Kankonian "Wantas sequence" for the 1, 2, 6, 12, 60, 420, 840 . . . sequence. Khemehekis (talk) 13:32, 21 December 2014 (UTC)[reply]

Kankonian appears to be Khemehekis conlang. —Tamfang (talk) 01:14, 22 December 2014 (UTC)[reply]

Numbers formed by concatenating consecutive integers would include your examples as well as numbers such as 91,011. A subset of that is OEIS:A007908 ("Concatenation of the numbers from 1 to n. "). -- ToE 21:24, 21 December 2014 (UTC)[reply]