ON THE INTERSECTION OF TWO DISTINCT k-GENERALIZED FIBONACCI SEQUENCES

Let k >= 2 and define F-(k) := (F-n((k))) n >= 0, the k-generalized Fibonacci sequence whose terms satisfy the recurrence relation F-n((k)) = F-n - 1((k)) + F ((k))(n - 2) +... + F ((k))(n - k) , with initial conditions 0, 0,..., 0, 1 (k terms) and such that the first nonzero term is F-1((k)) = 1. The sequences F := F ((2)) and T := F ((3)) are the known Fibonacci and Tribonacci sequences, respectively. In 2005, Noe and Post made a conjecture related to the possible solutions of the Diophantine equation F-n((k)) = F-m ((l)). In this note, we use transcendental tools to provide a general method for finding the intersections F-(k)boolean AND F-(m) which gives evidence supporting the Noe- Post conjecture. In particular, we prove that F boolean AND T = {0, 1, 2, 13}.