The probability that no two consecutive heads appear in n tosses of a coin is Fn+2 / 2n,
where Fn is the Fibonacci sequence, defined by the recurrence relation Fn = Fn−1 + Fn−2, for n > 2, with F1 = F2 = 1.
A closed form formula for the Fibonacci sequence is Fn = (Phin − phin)/,
where Phi = (1 + )/2 and phi = (1 − )/2 are the roots of the quadratic equation x2 − x − 1 = 0.
Therefore the above probability can also be written as (Phin+2 − phin+2) / 2n·.