A362149 -id:A362149 - cp's OEIS frontend

A362150 Decimal expansion of lambda, a constant arising in the analysis of the binary Euclidean algorithm.

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

Offset: 0

Paolo Xausa, Apr 09 2023

See Brent (1999), p. 12, and Morris (2016), p. 79, where this constant is called zeta(1).

See A362149 for additional comments, references and links.

			0.3979226811883166440767071611426549823098...

Richard P. Brent, Further analysis of the binary Euclidean algorithm, arXiv:1303.2772 [cs.DS], 1999, p. 12.
Ian D. Morris, A rigorous version of R. P. Brent's model for the binary Euclidean algorithm, Advances in Mathematics, Vol. 290, 26 Feb. 2016, p. 79.

Cf. A118858, A345987, A362149.

Equals (4*log(2)/Pi^2)/A362149 = 4*A118858/A362149.

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

A362150 Decimal expansion of lambda, a constant arising in the analysis of the binary Euclidean algorithm.

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