Feller's coin-tossing constants
Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in n independent tosses of a fair coin, no run of k consecutive heads (or, equally, tails) appears.
where αk is the smallest positive real root of
Values of the constants
For the constants are related to the golden ratio, , and Fibonacci numbers; the constants are and . The exact probability p(n,2) can be calculated either by using Fibonacci numbers, p(n,2) = or by solving a direct recurrence relation leading to the same result. For higher values of , the constants are related to generalizations of Fibonacci numbers such as the tribonacci and tetranacci numbers. The corresponding exact probabilities can be calculated as p(n,k) = . 
If we toss a fair coin ten times then the exact probability that no pair of heads come up in succession (i.e. n = 10 and k = 2) is p(10,2) = = 0.140625. The approximation gives 1.44721356...×1.23606797...−11 = 0.1406263...
- Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Volume 1 (3rd Edition), Wiley. ISBN 0-471-25708-7 Section XIII.7
- Quora Question: "You toss a fair coin 12 times. What is the probability that two heads do not occur consecutively?"
- Coin Tossing at WolframMathWorld