Abstract
Let (F n ) n≥0 be the Fibonacci sequence given by F n+2 = F n+1 + F n , for n ≥ 0, where F 0 = 0 and F 1 = 1. There are several interesting identities involving this sequence such as F 2 n + F 2 n+1 = F 2n+1, for all n ≥ 0. In a very recent paper, Marques and Togbé proved that if F s n + F s n+1 is a Fibonacci number for all sufficiently large n, then s = 1 or 2. In this paper, we will prove, in particular, that if (G m ) m is a linear recurrence sequence (under weak assumptions) and G s n + ... + G s n+k ∈ (G m ) m , for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on k and the parameters of G m .
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Scholarship holder of Capes-Brazil.
Supported by FEMAT and CNPq-Brazil.
Supported in part by Purdue University North Central.
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Chaves, A.P., Marques, D. & Togbé, A. On the sum of powers of terms of a linear recurrence sequence. Bull Braz Math Soc, New Series 43, 397–406 (2012). https://doi.org/10.1007/s00574-012-0018-y
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DOI: https://doi.org/10.1007/s00574-012-0018-y