On the spacing between terms of generalized Fibonacci sequences
Volume 134 / 2014
Colloquium Mathematicum 134 (2014), 267-280
MSC: Primary 11B39; Secondary 11J86.
DOI: 10.4064/cm134-2-10
Abstract
For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined to have the initial $k$ terms $0,0,\ldots ,0,1$ and be such that each term afterwards is the sum of the $k$ preceding terms. We will prove that the number of solutions of the Diophantine equation $F_m^{(k)}-F_n^{(\ell )}=c>0$ (under some weak assumptions) is bounded by an effectively computable constant depending only on $c$.
