cp's OEIS Frontend

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

%I A195596 #33 May 16 2023 12:11:37
%S A195596 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,
%T A195596 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,
%U A195596 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
%N A195596 Decimal expansion of alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.
%C A195596 alpha is used to measure the expected height of random binary search trees.
%D A195596 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.13 Binary search tree constants, p. 352.
%H A195596 Alois P. Heinz, <a href="/A195596/b195596.txt">Table of n, a(n) for n = 1..10000</a>
%H A195596 Luc Devroye, <a href="http://luc.devroye.org/devroye_1986_univ_a_note_on_the_height_of_binary_search_trees.pdf">A note on the height of binary search trees</a>, Journal of the ACM, Vol. 33, No. 3 (1986), pp. 489-498.
%H A195596 Bruce Reed, <a href="http://doi.acm.org/10.1145/765568.765571">The height of a random binary search tree</a>, J. ACM, 50 (2003), 306-332.
%H A195596 John Michael Robson, <a href="https://50years.acs.org.au/content/dam/acs/50-years/journals/acj/ACJ-V11-N04-197911.pdf#page=41">The height of binary search trees</a>, Australian Computer Journal, Vol. 11, No. 4 (1979), pp. 151-153. [broken link]
%H A195596 Larry Shepp, Doron Zeilberger and Cun-Hui Zhang, <a href="https://arxiv.org/abs/1210.5642">Pick up sticks</a>, arXiv preprint arXiv:1210.5642 [math.CO] (2012).
%H A195596 Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_search_tree">Binary search tree</a>
%H A195596 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%H A195596 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A195596 alpha = -1/W(-exp(-1)/2), where W is the Lambert W function.
%F A195596 A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1), with beta = 1.953... (A195599).
%e A195596 4.31107040700100503504707609644689027839156299804028805066937...
%p A195596 alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
%p A195596 as:= convert(evalf(alpha/10, 130), string):
%p A195596 seq(parse(as[n+1]), n=1..120);
%t A195596 RealDigits[ -1/ProductLog[-1/(2*E)] , 10, 105] // First (* _Jean-François Alcover_, Feb 19 2013 *)
%Y A195596 Cf. A195597 (continued fraction), A195598 (Engel expansion), A195581, A195582, A195583, A195599, A195600, A195601.
%K A195596 nonn,cons
%O A195596 1,1
%A A195596 _Alois P. Heinz_, Sep 21 2011