A195601 Engel expansion of beta = 3/(2*log(alpha/2)); alpha = A195596.

1, 2, 2, 2, 2, 5, 5, 5, 20, 36, 78, 842, 5291, 10373, 17340, 28619, 35586, 93572, 98045, 2470364, 13603654, 14328528, 16490766, 833971648, 1788088151, 9592330101, 10952282168, 40005288076, 54302548920, 118523737357, 776601533408, 1241894797770, 24485470725324

Offset: 1

Author

Alois P. Heinz, Sep 21 2011

Keywords

• nonn

Comments

beta = 1.95302570335815413945406288542575380414251340201036319609354... is used to measure the expected height of random binary search trees.

Cf. A006784 for definition of Engel expansion.

References

• F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
• P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
• B. Reed, The height of a random binary search tree, J. ACM, 50 (2003), 306-332.
• Eric Weisstein's World of Mathematics, Engel Expansion
• Wikipedia, Engel Expansion
• Index entries for sequences related to Engel expansions

Crossrefs

Cf. A195599 (decimal expansion), A195600 (continued fraction), A195581, A195582, A195583, A195596, A195597, A195598.

Programs

Maple

alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
beta:= 3/(2*log(alpha/2)):
engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
Digits:=400: engel(evalf(beta), 39);

Mathematica

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 *)

Formula

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.

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