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Continued fraction for beta = 3/(2*log(alpha/2)); alpha = A195596.
7

%I #17 Jul 03 2024 10:54:36

%S 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 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 12,1,7,3,2,26,3,2,1,1,1,9,1,15,4,3,3,1,3,1

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

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

%H 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 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 A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1).

%e 1.95302570335815413945406288542575380414251340201036319609354...

%p with(numtheory):

%p alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):

%p beta:= 3/(2*log(alpha/2)):

%p cfrac(evalf(beta, 130), 100, 'quotients')[];

%t beta = 3/(2+2*ProductLog[-1/(2*E)]); ContinuedFraction[beta, 83] (* _Jean-François Alcover_, Jun 20 2013 *)

%Y Cf. A195599 (decimal expansion), A195601 (Engel expansion), A195581, A195582, A195583, A195596, A195597, A195598.

%K nonn,cofr

%O 0,3

%A _Alois P. Heinz_, Sep 21 2011

%E Offset changed by _Andrew Howroyd_, Jul 03 2024