Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.8

On the Sum of Reciprocals of Numbers Satisfying a Recurrence Relation of Order s


Takao Komatsu
Graduate School of Science and Technology
Hirosaki University
Hirosaki, 036-8561
Japan

Vichian Laohakosol
Department of Mathematics
Kasetsart University
Bangkok 10900
Thailand

Abstract:

We discuss the partial infinite sum $ \sum_{k=n}^{\infty}u_k^{-s}$ for some positive integer $ n$, where $ u_k$ satisfies a recurrence relation of order $ s$, $ u_n= a u_{n-1}+u_{n-2}+\cdots+u_{n-s}$ ($ n\ge s$), with initial values $ u_0\ge 0$, $ u_k\in\mathbb{N}$ ( $ 0\le k\le s-1$), where $ a$ and $ s(\ge 2)$ are positive integers. If $ a=1$, $ s=2$, and $ u_0=0$, $ u_1=1$, then $ u_k=F_k$ is the $ k$-th Fibonacci number. Our results include some extensions of Ohtsuka and Nakamura. We also consider continued fraction expansions that include such infinite sums.

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(Concerned with sequence A000073.)

Received January 20 2010; revised version received May 19 2010. Published in Journal of Integer Sequences, May 20 2010.

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