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A093602 Decimal expansion of Pi/sqrt(3) = sqrt(2*zeta(2)). 13
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 (list; constant; graph; refs; listen; history; text; internal format)
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 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." - Jonathan Vos Post, Aug 10 2007

Equals integral_{x=0..infinity} x^(1/3)/(1+x^2). - Jean-Fran├žois Alcover, May 24 2013

Equals (3/2)*( integral_{x=0..infinity} 1/(1+x+x^2) dx ). - Bruno Berselli, Jul 23 2013

Volume of a cube with edge length 1 rotated about a space diagonal. See MathWorld Cube page. - Francis Wolinski, Mar 10 2019

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, Lieb-Thirring inequalities with improved constants, arXiv:0708.1165 [math.AP], 2007.

Eric Weisstein's World of Mathematics, No-Three-in-a-Line-Problem

Eric Weisstein's World of Mathematics, Cube

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

From Peter Bala, Nov 02 2019: (Start)

Pi/sqrt(3) = 3*Sum_{n >= 1}  1/( (3*n - 1)*(3*n - 2) ).

Pi/sqrt(3) = 2 - 6*Sum_{n >= 1}  1/( (3*n - 1)*(3*n + 1)*(3*n + 2) ).

Pi/sqrt(3) = 5!*Sum_{n >= 1}  1/( (3*n - 1)*(3*n - 2)*(3*n + 2)*(3*n + 4) ).

Pi/sqrt(3) = 3*( 1 - 2*Sum_{n >= 1} 1/(9*n^2 - 1) ).

Pi/sqrt(3) = 1 + Sum_{n >=1 } (-1)^(n+1)*(6*n + 1)/(n*(n + 1)*(3*n + 1)*(3*n - 2)).

Pi/sqrt(3) = (27/2)*Sum_{n >= 1} (2*n + 1)/( (3*n - 1)*(3*n + 1)*(3*n + 2)*(3*n + 4) ).

Pi/sqrt(3) = 3*Integral_{x = 0..1} 1/(1 + x + x^2) dx.

Pi/sqrt(3) = 3*Integral_{x = 0..1} (1 + x)/(1 - x + x^2) dx.

Pi/sqrt(3) = 3*Integral_{x = 0..inf} cosh(x)/cosh(3*x) dx.

(End)

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=(x-d)*10; write("b093602.txt", n, " ", d)); } \\ Harry J. Smith, Jun 19 2009

(MAGMA) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/Sqrt(3); // G. C. Greubel, Mar 10 2019

(Sage) numerical_approx(pi/sqrt(3), digits=100) # G. C. Greubel, Mar 10 2019

CROSSREFS

Cf. A132116.

Equals twice 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

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Last modified February 29 05:25 EST 2020. Contains 332353 sequences. (Running on oeis4.)