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The generalized order-\(k\) Lucas sequences in finite groups. (English) Zbl 1241.11013

Summary: We study the generalized order-\(k\) Lucas sequences modulo \(m\). Also, we define the \(i\)th generalized order-\(k\) Lucas orbit \(l^{i,\{\alpha_1,\alpha_2,\dots,\alpha_{k-1}\}}(G)\) with respect to the generating set \(A\) and the constants \(\alpha_1, \alpha_2\), and \(\alpha_{k-1}\) for a finite group \(G = \langle A \rangle\). Then, we obtain the lengths of the periods of the \(i\)th generalized order-\(k\) Lucas orbits of the binary polyhedral groups \(\langle n, 2, 2 \rangle, \langle 2, n, 2 \rangle, \langle 2, 2, n \rangle\) and the polyhedral groups \((n, 2, 2), (2, n, 2), (2, 2, n)\) for \(1 \leq i \leq k\).

MSC:

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