Testwiki:Reference desk/Archives/Mathematics/2020 October 8

{| width = "100%"

|- ! colspan="3" align="center" | Mathematics desk |- ! width="20%" align="left" | < October 7 ! width="25%" align="center"|<< Sep | October | Nov >> ! width="20%" align="right" |Current desk > |}

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

October 8

Which formula is correct??

For the centered hexagonal numbers 1, 7, 19, 37, 61, 91... I always thought that $3n^2-3n+1$ was the formula. Wikipedia's centered hexagonal number article agrees with me, but Mathworld and the OEIS say that it's $3n^2+3n+1$ . Who is right?? Georgia guy (talk) 14:46, 8 October 2020 (UTC)

They both are. These are the same sequences, just offset by 1 ($a_1=1$ vs $a_0=1$), so it's just a matter of convention where you start indexingTemplate:Sndkind of like how you can find the Fibonacci sequence indexed from 0 or 1. –Deacon Vorbis (carbon • videos) 15:11, 8 October 2020 (UTC)

All are correct, but same are more correct than others. The Fibonacci numbers satisfy the beautiful divisibility property that

$\gcd (F_m,F_n)=F_{\gcd (m,n)},$

but only if the indexing is such that $F_0=0.$ I do not know of a strong argument, though, that favours one of the two reasonable indexing choices for the centred hexagonal numbers.  --Lambiam 22:53, 8 October 2020 (UTC)

I also like $F_{12}=12^2$ pma 23:37, 13 October 2020 (UTC)