[seqfan] Re: Inconsistent offsets
Charles Greathouse
charles.greathouse at case.edu
Wed Dec 7 18:07:38 CET 2016
It sounds like the 28-gonal numbers need to be fixed, they should start
a(0) = 0, a(1) = 1 like the others.
I think of the centered polygonal numbers as starting at 1 (that's the
"center" of centered polygonal numbers). I would expect that these start at
a(1) = 1 but I don't have strong enough feelings here to change all of them
(though someone could).
Charles Greathouse
Case Western Reserve University
On Tue, Dec 6, 2016 at 6:06 AM, Daniel <kimpire at yahoo.com> wrote:
> 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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>
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