cp's OEIS Frontend

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

%I A195600 #17 Jul 03 2024 10:54:36
%S A195600 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,
%T A195600 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,
%U A195600 12,1,7,3,2,26,3,2,1,1,1,9,1,15,4,3,3,1,3,1
%N A195600 Continued fraction for beta = 3/(2*log(alpha/2)); alpha = A195596.
%C A195600 beta is used to measure the expected height of random binary search trees.
%H A195600 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 A195600 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.
%F A195600 A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1).
%e A195600 1.95302570335815413945406288542575380414251340201036319609354...
%p A195600 with(numtheory):
%p A195600 alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
%p A195600 beta:= 3/(2*log(alpha/2)):
%p A195600 cfrac(evalf(beta, 130), 100, 'quotients')[];
%t A195600 beta = 3/(2+2*ProductLog[-1/(2*E)]); ContinuedFraction[beta, 83] (* _Jean-François Alcover_, Jun 20 2013 *)
%Y A195600 Cf. A195599 (decimal expansion), A195601 (Engel expansion), A195581, A195582, A195583, A195596, A195597, A195598.
%K A195600 nonn,cofr
%O A195600 0,3
%A A195600 _Alois P. Heinz_, Sep 21 2011
%E A195600 Offset changed by _Andrew Howroyd_, Jul 03 2024