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A195598
Engel expansion of alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.
6
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
1,5
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
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).
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);
CROSSREFS
Cf. A195596 (decimal expansion), A195597 (continued fraction), A195581, A195582, A195583, A195599, A195600, A195601.
Sequence in context: A154857 A019314 A335820 * A120132 A331263 A334018
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 21 2011
STATUS
approved