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A195597
Continued fraction for alpha, the unique solution on [2,oo) of the equation alpha*log((2*e)/alpha)=1.
6
4, 3, 4, 1, 1, 1, 11, 2, 19, 1, 3, 1, 1, 1, 14, 1, 3, 5, 58, 3, 1, 10, 1, 1, 6, 5, 13, 127, 1, 1, 7, 13, 1, 2, 1, 2, 2, 1, 2, 2, 4, 2, 4, 1, 1, 6, 9, 3, 1, 16, 1, 3, 2, 32, 3, 1, 1, 2, 11, 1, 13, 4, 2, 1, 1, 1, 1, 2, 2, 6, 1, 1, 1, 2, 25, 1, 5, 5, 1, 1, 1, 1, 5, 2, 3, 2, 5, 25, 1, 190, 2, 1, 5, 3, 1, 20, 1, 1, 2, 1, 3
OFFSET
0,1
COMMENTS
alpha is used to measure the expected height of random binary search trees.
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).
EXAMPLE
4.31107040700100503504707609644689027839156299804028805066937...
MAPLE
with(numtheory):
alpha:= solve(alpha*log((2*exp(1))/alpha)=1, alpha):
cfrac(evalf(alpha, 130), 100, 'quotients')[];
MATHEMATICA
alpha = -1/ProductLog[-1/(2*E)]; ContinuedFraction[alpha, 101] (* Jean-François Alcover, Jun 20 2013 *)
CROSSREFS
Cf. A195596 (decimal expansion), A195598 (Engel expansion), A195581, A195582, A195583, A195599, A195600, A195601.
Sequence in context: A070431 A070511 A066340 * A143505 A245727 A280822
KEYWORD
nonn,cofr
AUTHOR
Alois P. Heinz, Sep 21 2011
EXTENSIONS
Offset changed by Andrew Howroyd, Jul 03 2024
STATUS
approved