The probability that no two consecutive heads appear in n tosses of a coin is F_{n+2} / 2^{n},

where F_{n} is the Fibonacci sequence, defined by the recurrence relation F_{n} = F_{n−1} + F_{n−2}, for n > 2, with F_{1} = F_{2} = 1.

A closed form formula for the Fibonacci sequence is F_{n} = (Phi^{n} − phi^{n})/,

where Phi = (1 + )/2 and phi = (1 − )/2 are the roots of the quadratic equation x^{2} − x − 1 = 0.

Therefore the above probability can also be written as (Phi^{n+2} − phi^{n+2}) / 2^{n}·.