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.

William Feller showed that if this probability is written as p(n,k) then

${\displaystyle \lim _{n\rightarrow \infty }p(n,k)\alpha _{k}^{n+1}=\beta _{k}}$

where α_k is the smallest positive real root of

${\displaystyle x^{k+1}=2^{k+1}(x-1)}$

and

${\displaystyle \beta _{k}={2-\alpha _{k} \over k+1-k\alpha _{k}}.}$