Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.5

A Self-Indexed Sequence

Emmanuel Preissmann
11 rue Vinet
1004 Lausanne

Abstract: We investigate the integer sequence $ \left(t_{n}\right)_{n\in\mathbb{Z}}$ defined by $ t_{n}=0$ if $ n\leq0$, $ t_{1}=1$, and $ t_{n}=\sum_{i=1}^{n-1}t_{n-t_{i}}$ for $ n \geq 2$. This sequence has the following properties: if we consider $ f_{n}(X):=-1+\sum_{i=1}^{n}X^{t_{i}}$ and take $ x_{n}$ to be the real positive number such that $ f_{n}(x_{n})=0$, then

$\displaystyle \lim_{n\rightarrow\infty}\frac{t_{n}}{t_{n+1}}=\lim_{n\rightarrow\infty}x_{n}=0.410098516\cdots$

Moreover, if $ u$ is the real positive number such that $ 1=\sum_{i=1}^{\infty}
u^{-t_i}$, then there is a positive constant $ M$ such that $ t_n\sim Mu^n$.

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(Concerned with sequences A052109 .)

Received May 16 2005; revised version received July 28 2005. Published in Journal of Integer Sequences August 2 2005.

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