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

1, 1, 1, 1, 4, 5, 5, 10, 15, 18, 102, 114, 246, 394, 1051, 3044, 50263, 111686, 128162, 273256, 583069, 927699, 7299350, 10833746, 15187876, 67314562, 2141820499, 4969978969, 10131201410, 49316153957, 221808008142, 275241196373, 1466049587038, 3406190692970

Offset: 1

Author

Alois P. Heinz, Sep 21 2011

Keywords

• nonn

Comments

alpha = 4.31107040700100503504707609644689027839156299804028805066937... 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

• 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. A195596 (decimal expansion), A195597 (continued fraction), A195581, A195582, A195583, A195599, A195600, A195601.

Programs

Maple

alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
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(alpha), 39);

Formula

alpha = -1/W(-exp(-1)/2), where W is the Lambert W function.

A195582(n)/A195583(n) = alpha*log(n) - beta*log(log(n)) + O(1), with beta = 1.953... (A195599).