Restricted Binary Strings and Generalized Fibonacci Numbers

We provide some interesting relations involving k-generalized Fibonacci numbers between the set F-n((k)) n of length n binary strings avoiding k of consecutive 0's and the set of length n strings avoiding k + 1 consecutive 0's and 1's with some more restriction on the first and last letter, via a simple bijection. In the special case k = 2 a probably new interpretation of Fibonacci numbers is given. Moreover, we describe in a combinatorial way the relation between the strings of F-n((k)) n with an odd numbers of 1's and the ones with an even number of 1's.