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Zbl 0866.11011
Dikici, R.; Smith, G.C.
Recurrences in finite groups. (English)
[J] Turk. J. Math. 19, No.3, 321-329 (1995). ISSN 1300-0098; ISSN 1303-6149/e

In a finite $p$-group $G$, let $(r_i)$ be a 3-step Fibonacci sequence defined by the recurrence $r_{i+3}=r_i+r_{i+1}+ r_{i+2}$ with given initial terms $r_0,r_1,r_2$. It is trivial that the sequence $(r_i)$ is periodic. Let $k(G)$ be the least common multiple of the fundamental periods of all sequences in $G$ satisfying the recurrence and denote by $k$ the fundamental period of the sequence when $r_0=r_1=0$, $r_2=1$. The main theorem of the paper is the following: ``Let $p>3$ be a prime number, then if $G$ is a non-trivial finite $p$-group of exponent $p$ and nilpotency class 2, then $k(G)=k$''.
[Péter Kiss (Eger)]

MSC 2000:: *11B39 Special numbers, etc.
20D60 Arithmetic and combinatorial problems on finite groups
20D15 Nilpotent finite groups

Keywords: recurrences; periodic sequence; finite $p$-group; 3-step Fibonacci sequence; fundamental periods

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