Coupled Fibonacci sequences of lower order have been generalized in number of ways.In this paper the Multiplicative Coupled Fibonacci Sequence has been generalized for r t h order with some new interesting properties. MSC: 11B39 • 11B37 • 11B99

For a finitely generated group G = 〈A〉 where A = {a1, a2, . . . , an} the sequence xi = ai+1, 0 ≤ i ≤ n − 1, xi+n = ∏n j=1 xi+j−1, i ≥ 0, is called the Fibonacci orbit of G with respect to the generating set A, denoted FA(G). If FA(G) is periodic, we call the length of the period of the sequence the Fibonacci length of G with respect to A, written LENA(G). We examine the Fibonacci lengths of al...

In this paper we present another characterization of (±1)-invariant sequences. We also introduce truncated Fibonacci and Lucas sequences of the second kind and show that a sequence x ∈ R∞ is (−1)-invariant(1-invariant resp.) if and only if D[ 0 x ] is perpendicular to every truncated Fibonacci(truncated Lucas resp.) sequence of the second kind where D = diag((−1), (−1), (−1), . . .).

Fibonacci cubes, extended Fibonacci cubes, and Lucas cubes are induced subgraphs of hypercubes 9 defined in terms of Fibonacci strings. It is shown that all these graphs are median. Several enumeration results on the number of their edges and squares are obtained. Some identities involving Fibonacci 11 and Lucas numbers are also presented. © 2005 Published by Elsevier B.V. 13

Let ftm = 0111010010001⋯ be the analogue of Thue-Morse sequence in Fibonacci representation. In this note we show how, using Walnut theorem-prover, to obtain exact value its maximum order complexity, previously studied by Jamet, Popoli, and Stoll. We strengthen one their theorems disprove conjectures.