A195599 - OEIS

%I #19 Jun 29 2020 18:11:22

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

%T 8,0,4,1,4,2,5,1,3,4,0,2,0,1,0,3,6,3,1,9,6,0,9,3,5,4,2,8,8,1,8,0,6,9,

%U 6,0,7,9,7,2,3,3,6,2,5,2,5,6,9,7,5,2,1,8,9,2,9,5,3,3,5,3,1,5,1,9,7,3,2,3,1

%N Decimal expansion of beta = 3/(2*log(alpha/2)), where alpha = A195596.

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

%H Alois P. Heinz, <a href="/A195599/b195599.txt">Table of n, a(n) for n = 1..10000</a>

%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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_search_tree">Binary search tree</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

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

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

%p bs:= convert(evalf(beta/10, 130), string):

%p seq(parse(bs[n+1]), n=1..120);

%t RealDigits[ 3/(2 + 2*ProductLog[-1/(2*E)]) , 10, 105] // First (* _Jean-François Alcover_, Feb 19 2013 *)

%Y Cf. A195600 (continued fraction), A195601 (Engel expansion), A195581, A195582, A195583, A195596, A195597, A195598.

%K nonn,cons

%O 1,2

%A _Alois P. Heinz_, Sep 21 2011