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

1,2

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

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).

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 *)

Crossrefs

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

Sequence in context: A288026 A141299 A035681 * A352723 A222255 A350970

Adjacent sequences: A195598 A195599 A195600 * A195602 A195603 A195604

Keyword

nonn

Author

Alois P. Heinz, Sep 21 2011

Status

approved