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A195598 Engel expansion of alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.

%I A195598 #30 Feb 16 2025 08:33:15
%S A195598 1,1,1,1,4,5,5,10,15,18,102,114,246,394,1051,3044,50263,111686,128162,
%T A195598 273256,583069,927699,7299350,10833746,15187876,67314562,2141820499,
%U A195598 4969978969,10131201410,49316153957,221808008142,275241196373,1466049587038,3406190692970
%N A195598 Engel expansion of alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.
%C A195598 alpha = 4.31107040700100503504707609644689027839156299804028805066937... is used to measure the expected height of random binary search trees.
%C A195598 Cf. A006784 for definition of Engel expansion.
%D A195598 F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.
%H A195598 F. Engel, <a href="/A006784/a006784.pdf">Entwicklung der Zahlen nach Stammbruechen</a>, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
%H A195598 P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
%H A195598 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 A195598 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H A195598 Wikipedia, <a href="https://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%H A195598 <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%F A195598 alpha = -1/W(-exp(-1)/2), where W is the Lambert W function.
%F A195598 A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1), with beta = 1.953... (A195599).
%p A195598 alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
%p A195598 engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
%p A195598 Digits:=400: engel(evalf(alpha), 39);
%Y A195598 Cf. A195596 (decimal expansion), A195597 (continued fraction), A195581, A195582, A195583, A195599, A195600, A195601.
%K A195598 nonn
%O A195598 1,5
%A A195598 _Alois P. Heinz_, Sep 21 2011