A195600 Continued fraction for beta = 3/(2*log(alpha/2)); alpha = A195596.

1, 1, 20, 3, 2, 7, 1, 1, 1, 12, 1, 5, 1, 91, 1, 1, 3, 87, 2, 1, 1, 1, 1, 3, 1, 9, 3, 2, 1, 1, 1, 1, 190, 1, 3, 1, 82, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 2, 12, 6, 2, 2, 2, 3, 2, 1, 1, 1, 2, 3, 21, 1, 1, 12, 1, 7, 3, 2, 26, 3, 2, 1, 1, 1, 9, 1, 15, 4, 3, 3, 1, 3, 1

Offset: 0

Author

Alois P. Heinz, Sep 21 2011

Keywords

• nonn
• cofr

Comments

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

Examples

			1.95302570335815413945406288542575380414251340201036319609354...
		

Links

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

Crossrefs

Cf. A195599 (decimal expansion), A195601 (Engel expansion), A195581, A195582, A195583, A195596, A195597, A195598.

Programs

Maple

with(numtheory):
alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
beta:= 3/(2*log(alpha/2)):
cfrac(evalf(beta, 130), 100, 'quotients')[];

Mathematica

beta = 3/(2+2*ProductLog[-1/(2*E)]); ContinuedFraction[beta, 83] (* Jean-François Alcover, Jun 20 2013 *)

Formula

beta = 3/(2*log(alpha/2)) = 3*alpha/(2*alpha-2), where alpha = A195596 = -1/W(-exp(-1)/2) and W is the Lambert W function.

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

Extensions

Offset changed by Andrew Howroyd, Jul 03 2024