Journal of Integer Sequences, Vol. 11 (2008), Article 08.3.6

How the Shift Parameter Affects the Behavior of a Family of Meta-Fibonacci Sequences

Mu Cai and Stephen M. Tanny
Department of Mathematics
University of Toronto
Toronto, Ontario M5S 2E4


We explore the effect of different values of the shift parameter $s$ on the behavior of the family of meta-Fibonacci sequences defined by the $k$-term recursion

\begin{displaymath}T_{s,k}(n) := \displaystyle\sum_{i=0}^{k-1} T_{s,k}(n - i - s - T_{s,k}(n - i - 1)), \quad n > s + k, \quad k \geq 2\end{displaymath}

with the $s+k$ initial conditions $T_{s,k}(n) = 1$ for $1 \leq n
\leq s + k$. We show that for any odd $k \geq 3$ and non-negative integer $s$ the values in the sequence $T_{s,k}(n)$ and $T_{0,k}(n)$ are essentially the same. The only differences in these sequences are that each power of $k$ occurs precisely $k+s$ times in $T_{s,k}(n)$ and $k$ times in $T_{0,k}(n)$. For even $k$ the frequency of $k^r$ in $T_{0,k}(n)$ depends upon $r$. We conjecture that for $k$ even the effect of the shift parameter $s$ is analogous to that for $k$ odd, in the sense that the only differences in the sequences $T_{s,k}(n)$ and $T_{0,k}(n)$ occur in the frequencies of the powers of $k$; specifically, each power of $k$ appears to occur precisely $s$ more times in $T_{s,k}(n)$ than it does in $T_{0,k}(n)$.

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Received June 2 2008; revised version received August 15 2008. Published in Journal of Integer Sequences, August 17 2008.

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