k-generalized Fibonacci numbers of the form 1+2^{n_1}+4^{n_2}+\cdots+(2^{k})^{n_k}

Abstract

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence (F_n^{(k)})_{n>= 2-k} whose first k terms are 0, ..., 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, we investigate k-generalized Fibonacci numbers written in the form 1+2^{n_1}+4^{n_2}+\cdots+(2^{k})^{n_k}, for non-negative integers n_i, with n_k >= max{ n_i | 1<= i <= k-1}.