[seqfan] Re: Connection between A2450 and A321873?

[jean-paul allouche]

Hi

Both sequences involve Sum_{k>=1} 3/(4^k - 1)
Thus the only question is to justify the claim:


Sum_{n>=1} 1/a(n) converges to (3*(log[4/3]-QPolyGamma[0,1,1/4]))/log[4] =
1.26329305810027133188797266393139146884324008972340089723642183177... 
-K. G. Stier <http://oeis.org/wiki/User:K._G._Stier>, Jun 23 2014

Series similar to Sum_{k<=1} 3/(4^k-1) can be found, e.g., in
Borwein, J. M. and Borwein, P. B. "Evaluation of Sums of Reciprocals
of Fibonacci Sequences." §3.7 in Pi & the AGM: A Study in Analytic
Number Theory and Computational Complexity. New York: Wiley, pp. 91-101, 
1987,
where they are expressed in terms of a theta function. It remains to prove
the link with the PolyGamma function

best wishes
jp



Le 23/10/2019 à 06:30, Alonso Del Arte a écrit :
> One of the comments in A002450 states that
>
>> Sum_{n >= 1} 1/a(n) converges to (3*(log[4/3] - QPolyGamma[0, 1,
> 1/4]))/log[4] = 1.263293...
>
> Is this number the constant in A321873?
>
> Al
>