[seqfan] Re: A008441 Number of ways of writing n as the sum of 2 triangular numbers
Richard Mathar
mathar at strw.leidenuniv.nl
Thu Apr 7 16:25:08 CEST 2011
gg> Date: Thu, 7 Apr 2011 16:49:12 +0300
gg> From: Georgi Guninski
gg> To: seqfan at list.seqfan.eu
gg> Subject: [seqfan] A008441 Number of ways of writing n as the sum of 2 triangular numbers
gg>
gg> Since a(n) = A002654(4n+1) it will give some information about the divisors of $4n+1$ assuming just n is factored.
The formula in A008441 says
a(n)=b(4n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos Sep 14 2005
Since a(n)=A002654(4n+1) one would think that b(n)=A002654(n), but
then the formula argues that b(2^e)=0^e , whereas in A002654 we have
correctly the statement that a(2^e)=1. So the multiple choice question is:
i) is the Somos formula referring to b(n) = A002654(n)?
If yes, it should just be deleted, assuming that this is duplicate of information
and that the formula is wrong for p=2 and e>1.
ii) is the Somos formula referring to another multiplicative b(n) and
which one is this? (This should start
1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0
with offset 1 and appears to to A008442 or perhaps A126812 or perhaps both.
Richard J. Mathar
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