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On the periods of 2-step general Fibonacci sequences in dihedral groups. (English) Zbl 1119.11014

A group \(D_n\) is called dihedral if \(D_n=\langle a,b:a^n=e,\;b^2=e,\;ab=ba^{-1}\rangle\). The \(2\)-step general Fibonacci sequences in \(D_n\) are defined by \(x_0=a\), \(x_1=b\), \(x_i=x_{i-2}^m x_{i-1}^l\) (\(i\geq2\)) for integers \(m\) and \(l\). The authors consider the conditions where the \(2\)-step general Fibonacci sequences in \(D_n\) are simply periodic, namely have repetitions of fixed subsequences from the initial element \(a\). They also give the period of the sequences in such cases.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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