k-Lucas numbers and associated bipartite graphs

For a positive integer k greater than or equal to 2, the k-Fibonacci sequence {g(n)((k))} is defined as: g(1)((k)) = ... = g(k-2)((k)) = 0, g(k-1)((k)) = g(k)((k)) = 1 and for n > k greater than or equal to 2, g(n)((k)) = g(n-1)((k)) + g(n-2)((k)) + ... + g(n-k)((k)). Moreover, the k-Lucas sequence {l(n)((k))} is defined as l(n)((k)) = g(n-1)((k)) + g(n+k-1)((k)) for n greater than or equal to 1. In this paper we consider the relationship between g(n)((k)) and l(n)((k)) and 1-factor of a bipartite graph. (C) 2000 Elsevier Science Inc. All rights reserved.