A195597 Continued fraction for alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.

4, 3, 4, 1, 1, 1, 11, 2, 19, 1, 3, 1, 1, 1, 14, 1, 3, 5, 58, 3, 1, 10, 1, 1, 6, 5, 13, 127, 1, 1, 7, 13, 1, 2, 1, 2, 2, 1, 2, 2, 4, 2, 4, 1, 1, 6, 9, 3, 1, 16, 1, 3, 2, 32, 3, 1, 1, 2, 11, 1, 13, 4, 2, 1, 1, 1, 1, 2, 2, 6, 1, 1, 1, 2, 25, 1, 5, 5, 1, 1, 1, 1, 5, 2, 3, 2, 5, 25, 1, 190, 2, 1, 5, 3, 1, 20, 1, 1, 2, 1, 3

Offset: 0

Author

Alois P. Heinz, Sep 21 2011

Keywords

• nonn
• cofr

Comments

alpha is used to measure the expected height of random binary search trees.

Examples

			4.31107040700100503504707609644689027839156299804028805066937...
		

Links

• B. Reed, The height of a random binary search tree, J. ACM, 50 (2003), 306-332.

Crossrefs

Cf. A195596 (decimal expansion), A195598 (Engel expansion), A195581, A195582, A195583, A195599, A195600, A195601.

Programs

Maple

with(numtheory):
alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
cfrac(evalf(alpha, 130), 100, 'quotients')[];

Mathematica

alpha = -1/ProductLog[-1/(2*E)]; ContinuedFraction[alpha, 101] (* Jean-François Alcover, Jun 20 2013 *)

Formula

alpha = -1/W(-exp(-1)/2), where W is the Lambert W function.

A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1), with beta = 1.953... (A195599).

Extensions

Offset changed by Andrew Howroyd, Jul 03 2024