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

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

Offset

0,1

Comments

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.

Links

Table of n, a(n) for n=0..19.

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.

Formula

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).

Example

0.51606240889999180681...

Crossrefs

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

Sequence in context: A105060 A229096 A290797 * A200423 A176320 A190185

Adjacent sequences: A375063 A375064 A375065 * A375067 A375068 A375069

Keyword

nonn,cons,hard,more

Author

Paolo Xausa, Jul 29 2024

Status

approved