A174606 -id:A174606 - cp's OEIS frontend

A100199 Decimal expansion of Pi^2/(12*log(2)), inverse of Levy's constant.

1, 1, 8, 6, 5, 6, 9, 1, 1, 0, 4, 1, 5, 6, 2, 5, 4, 5, 2, 8, 2, 1, 7, 2, 2, 9, 7, 5, 9, 4, 7, 2, 3, 7, 1, 2, 0, 5, 6, 8, 3, 5, 6, 5, 3, 6, 4, 7, 2, 0, 5, 4, 3, 3, 5, 9, 5, 4, 2, 5, 4, 2, 9, 8, 6, 5, 2, 8, 0, 9, 6, 3, 2, 0, 5, 6, 2, 5, 4, 4, 4, 3, 3, 0, 0, 3, 4, 8, 3, 0, 1, 1, 0, 8, 4, 8, 6, 8, 7, 5, 9, 4, 6, 6, 3

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Dec 27 2004

From A.H.M. Smeets, Jun 12 2018: (Start)
The denominator of the k-th convergent obtained from a continued fraction of a constant, the terms of the continued fraction satisfying the Gauss-Kuzmin distribution, will tend to exp(k*A100199).
Similarly, the error between the k-th convergent obtained from a continued fraction of a constant, and the constant itself will tend to exp(-2*k*A100199). (End)
The term "Lévy's constant" is sometimes used to refer to this constant (Wikipedia). - Bernard Schott, Sep 01 2022

			1.1865691104156254528217229759472371205683565364720543359542542986528...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.7, p. 54.

G. C. Greubel, Table of n, a(n) for n = 1..10000
R. M. Corless, Continued Fractions and Chaos, Amer. Math. Monthly 99, 203-215, 1992.
Eric Weisstein's World of Mathematics, Khinchin-Levy Constant.
Eric Weisstein's World of Mathematics, Lévy Constant.
Wikipedia, Lévy's constant.

Cf. A086702, A089729, A072691, A002162, A174606.

RealDigits[Pi^2/(12*Log[2]), 10, 100][[1]] (* G. C. Greubel, Mar 23 2017 *)

Pi^2/log(4096) \\ Charles R Greathouse IV, Aug 04 2016

Equals 1/A089729 = log(A086702) = A174606/2.
Equals ((Pi^2)/12)/log(2) = A072691 / A002162 = (Sum_{n>=1} ((-1)^(n+1))/n^2) / (Sum_{n>=1} ((-1)^(n+1))/n^1). - Terry D. Grant, Aug 03 2016
Equals (-1/log(2)) * Integral_{x=0..1} log(x)/(1+x) dx (from Corless, 1992). - Bernard Schott, Sep 01 2022

A375066 Decimal expansion of Hensley's constant, arising in the analysis of the Euclidean algorithm.

Original entry on oeis.org

5, 1, 6, 0, 6, 2, 4, 0, 8, 8, 9, 9, 9, 9, 1, 8, 0, 6, 8, 1

Offset: 0

Paolo Xausa, Jul 29 2024

Appears in the formula for the asymptotic variance of the Euclidean algorithm.

When applying the Euclidean algorithm on pairs (a, b), with 0 <= a <= b <= x, the asymptotic formula for the variance of the number of steps (divisions), as x -> infinity, is H*log(x), where H is this constant. See Lhote (2004), eq. 1.8.

			0.51606240889999180681...

Philippe Flajolet and Brigitte Vallée, Continued Fractions, Comparison Algorithms, and Fine Structure Constants, Research Report RR-4072, INRIA, 2000, p. 24.
Doug Hensley, The Number of Steps in the Euclidean Algorithm, Journal of Number Theory, Vol. 49, Issue 2, November 1994, pp. 142-182.
Doug Hensley, Continued Fractions, World Scientific, 2006, p. 210.
Loïck Lhote, Computation of a class of continued fraction constants, in Arge et al., Proceedings of the Sixth Workshop on Algorithm Engineering and Experiments and the First Workshop on Analytic Algorithmics and Combinatorics, 2004, pp. 199-210.
Wikipedia, Euclidean algorithm.

Cf. A089729, A174606, A345987, A362149, A362150.

Equals 2*(lambda''(1) - lambda'(1)^2) / (-lambda'(1)^3), where lambda'(1) = -Pi^2/(6*log(2)) = -A174606 and lambda''(1) is 9.08037... See Lhote (2004), eq. 1.8, and Flajolet and Vallée (2000), p. 24 (where lambda''(1) is called the Hensley's constant).

cp's OEIS Frontend

A100199 Decimal expansion of Pi^2/(12*log(2)), inverse of Levy's constant.

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