# [seqfan] Re: Concerning A112299.

RGWv rgwv at rgwv.com
Sun Apr 3 00:54:34 CEST 2011

```Richard,

Thank you. That makes perfect sense.

Bob.

-----Original Message-----
From: Richard Mathar
Sent: Saturday, April 02, 2011 6:34 PM
To: seqfan at seqfan.eu
Subject: [seqfan] Re: Concerning A112299.

Back to
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

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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