We provide some interesting relations involving k-generalized Fibonacci numbers between the set $F_n^{(k)}$ 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 $F_n^{(k)}$ with an odd numbers of 1’s and the ones with an even number of 1’s.