A195599 Decimal expansion of beta = 3/(2*log(alpha/2)), where alpha = A195596.

1, 9, 5, 3, 0, 2, 5, 7, 0, 3, 3, 5, 8, 1, 5, 4, 1, 3, 9, 4, 5, 4, 0, 6, 2, 8, 8, 5, 4, 2, 5, 7, 5, 3, 8, 0, 4, 1, 4, 2, 5, 1, 3, 4, 0, 2, 0, 1, 0, 3, 6, 3, 1, 9, 6, 0, 9, 3, 5, 4, 2, 8, 8, 1, 8, 0, 6, 9, 6, 0, 7, 9, 7, 2, 3, 3, 6, 2, 5, 2, 5, 6, 9, 7, 5, 2, 1, 8, 9, 2, 9, 5, 3, 3, 5, 3, 1, 5, 1, 9, 7, 3, 2, 3, 1

Offset: 1

Author

Alois P. Heinz, Sep 21 2011

Keywords

• nonn
• cons

Comments

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

Examples

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Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• B. Reed, The height of a random binary search tree, J. ACM, 50 (2003), 306-332.
• Wikipedia, Binary search tree
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A195600 (continued fraction), A195601 (Engel expansion), A195581, A195582, A195583, A195596, A195597, A195598.

Programs

Maple

alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
beta:= 3/(2*log(alpha/2)):
bs:= convert(evalf(beta/10, 130), string):
seq(parse(bs[n+1]), n=1..120);

Mathematica

RealDigits[ 3/(2 + 2*ProductLog[-1/(2*E)]) , 10, 105] // First (* Jean-François Alcover, Feb 19 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).