Counting palindromic binary strings without r-runs of ones

A closed-form expression is derived for the enumeration of all palindromic binary strings of length n > r having no r-runs of 1’s, in terms of the r-Fibonacci sequence. A similar closed-form expression for the number of zeros contained in all such palindromic binary strings is derived in terms of the number of zeros contained in all binary strings having no r-runs of 1’s.