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Fibonacci sequences in finite groups. (English) Zbl 0758.20006

A \(k\)-nacci sequence in a finite group \(G\) is a sequence of group elements \(x_ 0,x_ 1,\dots,x_ n,\dots\) for which, given an initial generating set \(x_ 0,\dots,x_{j-1}\) for \(G\), each element is defined by \[ x_ n=\begin{cases} x_ 0x_ 1\cdots x_{n-1}&\text{for \(j\leq n<k\)}\\x_{n-k}x_{n-k+1}\cdots x_{n-1}&\text{for \(n\geq k\)}.\end{cases} \] 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.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
20F05 Generators, relations, and presentations of groups
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