A132116 Continued fraction expansion of Pi/sqrt(3).

1, 1, 4, 2, 1, 2, 3, 7, 3, 3, 30, 2, 1, 2, 2, 83, 9, 20, 1, 37, 1, 2, 7, 1, 1, 2, 1, 6, 1, 2, 1, 1, 3, 3, 1, 4, 8, 1, 6, 33, 1, 1, 1, 17, 4, 1, 3, 1, 5, 3, 2, 1, 1100, 2, 31, 6, 7, 1, 1, 9, 6, 3, 1, 2, 2, 2, 1, 2, 4, 6, 16, 1, 1, 8, 1, 13, 2, 18, 1, 4, 1, 46, 2, 5, 1, 3, 1, 42, 1, 1, 1, 26, 3, 2, 1, 5, 4

Offset: 0

Author

Jonathan Vos Post, Aug 10 2007

Keywords

• cofr
• easy
• nonn

Comments

Dolbeault et al. Abstract, where this is referred to as "the semiclassical constant" following remark 2, p. 2: "Following Eden and Foias we obtain a matrix version of a generalized Sobolev inequality in one-dimension. This allow us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schroedinger operators."

The inverse, sqrt(3)/Pi, which has the same continued fraction expansion (up to an initial zero), appears in geometric considerations involving spheres, see for example A343235. - M. F. Hasler, Oct 29 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 0..9999
• Jean Dolbeault, Ari Laptev and Michael Loss, Lieb-Thirring inequalities with improved constants, arXiv:0708.1165 [math.AP], 2007.

Crossrefs

Cf. A093602 (decimal expansion), A343235 (sqrt(3)/Pi - 0.5).

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(Pi(R)/Sqrt(3)); // G. C. Greubel, Sep 27 2018

Maple

with(numtheory): cfrac(Pi/(sqrt(3)),100,'quotients'); # Muniru A Asiru, Sep 28 2018

Mathematica

ContinuedFraction[Pi/Sqrt[3], 100] (* G. C. Greubel, Sep 27 2018 *)

PARI

default(realprecision, 100); contfrac(Pi/sqrt(3)) \\ G. C. Greubel, Sep 27 2018

Extensions

Offset changed by Andrew Howroyd, Aug 09 2024