cp's OEIS Frontend

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

%I A195597 #17 Jul 03 2024 10:54:32
%S A195597 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,
%T A195597 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,
%U A195597 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
%N A195597 Continued fraction for alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.
%C A195597 alpha is used to measure the expected height of random binary search trees.
%H A195597 B. 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.
%F A195597 alpha = -1/W(-exp(-1)/2), where W is the Lambert W function.
%F A195597 A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1), with beta = 1.953... (A195599).
%e A195597 4.31107040700100503504707609644689027839156299804028805066937...
%p A195597 with(numtheory):
%p A195597 alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
%p A195597 cfrac(evalf(alpha, 130), 100, 'quotients')[];
%t A195597 alpha = -1/ProductLog[-1/(2*E)]; ContinuedFraction[alpha, 101] (* _Jean-François Alcover_, Jun 20 2013 *)
%Y A195597 Cf. A195596 (decimal expansion), A195598 (Engel expansion), A195581, A195582, A195583, A195599, A195600, A195601.
%K A195597 nonn,cofr
%O A195597 0,1
%A A195597 _Alois P. Heinz_, Sep 21 2011
%E A195597 Offset changed by _Andrew Howroyd_, Jul 03 2024