On waiting time distribution of runs of ones or zeros in a Bernoulli sequence

Consider an infinite sequence of Bernoulli trials {Xi|i=1,2,…}{Xi|i=1,2,…}. Let W(k) denote the waiting time, the number of trials needed, to get either consecutive k ones or k zeros for the first time. The probability distribution of W(k) is derived for both independent and homogeneous two-state Markovian Bernoulli trials, using a generalized Fibonacci sequence of order kk. For independent Bernoulli trials, a special case of symmetric trial with p=12 is considered.