

A093602


Decimal expansion of Pi/sqrt(3) = sqrt(2*zeta(2)).


11



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

1,2


COMMENTS

Continued fraction expansion is A132116.  Jonathan Vos Post, Aug 10 2007
From Dolbeault et al.'s 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 generalised Sobolev inequality in onedimension. This allow us to improve on the known estimates of best constants in LiebThirring inequalities for the sum of the negative eigenvalues for multidimensional Schroedinger operators."  Jonathan Vos Post, Aug 10 2007
Equals integral_{x=0..infinity} x^(1/3)/(1+x^2).  JeanFrançois Alcover, May 24 2013
Equals (3/2)*( integral_{x=0..infinity} 1/(1+x+x^2) dx ).  Bruno Berselli, Jul 23 2013


LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20000
P. Bala, New series for old functions
Jean Dolbeault, Ari Laptev and Michael Loss, LiebThirring inequalities with improved constants, arXiv:0708.1165 [math.AP], 2007.
Eric Weisstein's World of Mathematics, NoThreeinaLineProblem


FORMULA

Pi/sqrt(3) = Sum_{n >=0} (1/(6*n+1)  4/(6*n+2)  5/(6*n+3)  1/(6*n+4) + 4/(6*n+5) + 5/(6*n+6)).  Mats Granvik, Sep 23 2013
Pi/sqrt(3) = (1/2) * Sum_{n >= 0} (14*n + 11)*(1/3)^n/((4*n + 1)*(4*n + 3)*binomial(4*n,2*n)). For more series representations of this type see the Bala link.  Peter Bala, Feb 04 2015


EXAMPLE

Pi/sqrt(3)=1.8137993642342178505940782576421557322840662480927405755...


MATHEMATICA

RealDigits[Pi/Sqrt[3], 10, 120][[1]] (* Harvey P. Dale, Mar 04 2012 *)


PROG

(PARI) { default(realprecision, 20080); x=Pi*sqrt(3)/3; for (n=1, 20000, d=floor(x); x=(xd)*10; write("b093602.txt", n, " ", d)); } \\ Harry J. Smith, Jun 19 2009


CROSSREFS

Cf. A132116. 2 * A093766.
Sequence in context: A092515 A193032 A127454 * A011469 A140457 A176457
Adjacent sequences: A093599 A093600 A093601 * A093603 A093604 A093605


KEYWORD

easy,nonn,cons


AUTHOR

Lekraj Beedassy, May 14 2004


STATUS

approved



