A195596 Decimal expansion of alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.

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

Offset: 1

Author

Alois P. Heinz, Sep 21 2011

Keywords

• nonn
• cons

Comments

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

Examples

			4.31107040700100503504707609644689027839156299804028805066937...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.13 Binary search tree constants, p. 352.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Luc Devroye, A note on the height of binary search trees, Journal of the ACM, Vol. 33, No. 3 (1986), pp. 489-498.
• Bruce Reed, The height of a random binary search tree, J. ACM, 50 (2003), 306-332.
• John Michael Robson, The height of binary search trees, Australian Computer Journal, Vol. 11, No. 4 (1979), pp. 151-153. [broken link]
• Larry Shepp, Doron Zeilberger and Cun-Hui Zhang, Pick up sticks, arXiv preprint arXiv:1210.5642 [math.CO] (2012).
• Wikipedia, Binary search tree
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees

Crossrefs

Cf. A195597 (continued fraction), A195598 (Engel expansion), A195581, A195582, A195583, A195599, A195600, A195601.

Programs

Maple

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

Mathematica

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