A132116 - OEIS

%I #34 Oct 30 2024 13:54:06

%S 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,

%T 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,

%U 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

%N Continued fraction expansion of Pi/sqrt(3).

%C 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."

%C 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

%H G. C. Greubel, <a href="/A132116/b132116.txt">Table of n, a(n) for n = 0..9999</a>

%H Jean Dolbeault, Ari Laptev and Michael Loss, <a href="https://arxiv.org/abs/0708.1165">Lieb-Thirring inequalities with improved constants</a>, arXiv:0708.1165 [math.AP], 2007.

%p with(numtheory): cfrac(Pi/(sqrt(3)),100,'quotients'); # _Muniru A Asiru_, Sep 28 2018

%t ContinuedFraction[Pi/Sqrt[3], 100] (* _G. C. Greubel_, Sep 27 2018 *)

%o (PARI) default(realprecision, 100); contfrac(Pi/sqrt(3)) \\ _G. C. Greubel_, Sep 27 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); ContinuedFraction(Pi(R)/Sqrt(3)); // _G. C. Greubel_, Sep 27 2018

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

%K cofr,easy,nonn

%O 0,3

%A _Jonathan Vos Post_, Aug 10 2007

%E Offset changed by _Andrew Howroyd_, Aug 09 2024