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

where α_k is the smallest positive real root of

and

For ${\displaystyle k=2}$ the constants are related to the golden ratio, ${\displaystyle \varphi }$, and Fibonacci numbers; the constants are ${\displaystyle {\sqrt {5}}-1=2\varphi -2=2/\varphi }$ and ${\displaystyle 1+1/{\sqrt {5}}}$. The exact probability p(n,2) can be calculated either by using Fibonacci numbers, p(n,2) = ${\displaystyle {\tfrac {F_{n+2}}{2^{n}}}}$ or by solving a direct recurrence relation leading to the same result. For higher values of ${\displaystyle k}$, 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) = ${\displaystyle {\tfrac {F_{n+2}^{(k)}}{2^{n}}}}$.

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