[seqfan] Re: Concerning A112299.
Richard Mathar
mathar at strw.leidenuniv.nl
Sun Apr 3 00:34:18 CEST 2011
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http://list.seqfan.eu/pipermail/seqfan/2011-April/014753.html
The value of sum_{k>=1} a(n)/n is the Dirichlet zeta-value of the sequence
at s=1. The sequence is periodic with period 8 (as the definition shows),
and a sum of 2 Dirichlet L-functions. The first one is the non-principal
A101455 that I called chi(r=4,m=8) in arXiv:1008.2547:
k= 1 2 3 4 5 6 7 8 ...(periodic)
1st 1 0 -1 0 1 0 -1 0
2nd 0 -1 0 0 0 1 0 0
------------------------------
a(n) 1 -1 -1 0 1 1 -1 0
The sum of the first is already given in A101455 (Pi/4).
The second is sum(-1/2+1/6-1/10+1/16 -... periodic with period 8 in denom)
= (-1/2)*sum(1-1/3+1/5-1/8 -... periodic with period 4 in denom)
= (-1/2)*sum(first) = (-1/2)*Pi/4
The sum is pi/4 -Pi/8 = pi/8.
This type of connections with arithmetic functions is the main reason
why some sequences start with a(1)=1 and offset 1.
Note that the two generating functions are the same after elimination
of common factors. I've added the factor x that I missed with the previous
editing.
RJM
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