[seqfan] Inconsistent offsets

Daniel kimpire at yahoo.com
Tue Dec 6 12:06:01 CET 2016


Hi all,

(Quick note about terminology before I begin: The OEIS uses n for the term "n-gon" as well as for noting where we are in a sequence, such as "n" and "a(n)". To avoid confusion about which "n" is which, I'm using "x-gon" instead of "n-gon" in this email.)

I've noticed an inconsistency in the offsets that we use for centered polygonal numbers.

Almost all of the non-centered x-gon sequences (A000290, A000326, A000566, A000567, A001106, etc.) start with the following (n, a(n)) pairings:
0,0
1,1
2,x
The one exception is A161935 (28-gonal numbers), which for some reason starts with (0,1) and (1,28).

The centered polygonal numbers are of two minds. For example:
* A005448, A069099, A060544, and A062786 begin with (1,1) and (2,x)
* A001844, A005891, A003215, and A016754 begin with (0,1) and (1,x)
I don't see any regularity about which ones start at n=0 and which start at n=1. I do notice, however, that none of them seem to start with a(n)=0.


So I have four questions:
1) I know we aren't supposed to worry too much about offsets as a general rule, but should we fix A161935 to bring it in line with all of the other x-gonal numbers?
2) If so, is it also worth editing half of the centered polygonal numbers to make them consistent with the other half?
3) If so, which half should we change?
4) Regardless of the answers to the above three questions, I'm curious: is there some fundamental reason that, as opposed to non-centered polygons, the centered polygons don't start with (0,0)?

Thanks,
Daniel Sterman


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