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%I #32 Nov 22 2020 17:36:47
%S 1,2,2,2,2,5,5,5,20,36,78,842,5291,10373,17340,28619,35586,93572,
%T 98045,2470364,13603654,14328528,16490766,833971648,1788088151,
%U 9592330101,10952282168,40005288076,54302548920,118523737357,776601533408,1241894797770,24485470725324
%N Engel expansion of beta = 3/(2*log(alpha/2)); alpha = A195596.
%C beta = 1.95302570335815413945406288542575380414251340201036319609354... is used to measure the expected height of random binary search trees.
%C Cf. A006784 for definition of Engel expansion.
%D F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.
%H G. C. Greubel, <a href="/A195601/b195601.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</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).
%p alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
%p beta:= 3/(2*log(alpha/2)):
%p engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
%p Digits:=400: engel(evalf(beta), 39);
%t f:= N[-1/ProductLog[-1/(2*E)], 500001]; EngelExp[A_, n_]:= Join[Array[1 &, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; EngelExp[N[3/(2*Log[f/2]), 500000], 25] (* _G. C. Greubel_, Oct 21 2016 *)
%Y Cf. A195599 (decimal expansion), A195600 (continued fraction), A195581, A195582, A195583, A195596, A195597, A195598.
%K nonn
%O 1,2
%A _Alois P. Heinz_, Sep 21 2011
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