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Zbl 0758.20006
Knox, Steven W.
Fibonacci sequences in finite groups.
(English)
[J] Fibonacci Q. 30, No.2, 116-120 (1992). ISSN 0015-0517

A $k$-nacci sequence in a finite group $G$ is a sequence of group elements $x\sb 0,x\sb 1,\dots,x\sb n,\dots$ for which, given an initial generating set $x\sb 0,\dots,x\sb{j-1}$ for $G$, each element is defined by $$x\sb n=\cases x\sb 0x\sb 1\cdots x\sb{n-1}&\text{for j\le n<k}\\x\sb{n-k}x\sb{n-k+1}\cdots x\sb{n-1}&\text{for n\ge k}.\endcases$$ A $k$-nacci sequence certainly reflects the structure of $G$. A finite group $G$ is called $k$-nacci sequenceable if there exists a $k$-nacci sequence of $G$ such that every element of $G$ appears in the sequence. It is shown that a $k$-nacci sequence in a finite group $G$ is simply periodic. This leads to a complete description of the 2-nacci sequenceable groups. A 2-nacci sequenceable group is cyclic.
[G.Rosenberger (Dortmund)]
MSC 2000:
*20D60 Arithmetic and combinatorial problems on finite groups
11B39 Special numbers, etc.
20F05 Presentations of groups

Keywords: generating set; $k$-nacci sequence; $k$-nacci sequenceable; finite group; 2-nacci sequenceable groups

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