Journal of Integer Sequences, Vol. 10 (2007), Article 07.10.2

On Fibonacci-Like Sequences


Le Anh Vinh
School of Mathematics
University of New South Wales
Sydney 2052 NSW
Australia

Abstract:

In this note, we study Fibonacci-like sequences that are defined by the recurrence $ S_k = a$, $ S_{k + 1} = b$, $ S_{n + 2} \equiv S_{n + 1} + S_n$ (mod $ n + 2$) for all $ n \geq k$, where $ k, a, b \in \mathbb{N}$, $ 0 \leq a < k$, $ 0 \leq b < k+ 1$, and $ (a,b)\ne (0,0)$. We will show that the number $ \alpha = 0.S_k S_{k+1} S_{k+2} \cdots$ is irrational. We also propose a conjecture on the pattern of the sequence $ \{S_n\}_{n \geq k}$.

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(Concerned with sequence A056542.)

Received October 21 2005; revised version received November 28 2007. Published in Journal of Integer Sequences, November 28 2007.

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