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

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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 indexing – kind 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)