A093602 Decimal expansion of Pi/sqrt(3) = sqrt(2*zeta(2)).

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

Offset: 1

Author

Lekraj Beedassy, May 14 2004

Keywords

• easy
• nonn
• cons

Comments

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

Volume of a cone with unit radius and 60-degree opening angle, and so height sqrt(3). Equivalently, the volume of the cone formed by rotating a 30-60-90 degree triangle with unit short leg about the long leg. - Christoph B. Kassir, Sep 17 2022

Examples

			Pi/sqrt(3) = 1.8137993642342178505940782576421557322840662480927405755...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• Peter Bala, New series for old functions
• Jean Dolbeault, Ari Laptev and Michael Loss, Lieb-Thirring inequalities with improved constants, Vol. 10, No. 4 (2008), pp. 1121-1126, preprint, arXiv:0708.1165 [math.AP], 2007.
• Sandi Klavžar, James Tuite, and Ullas Chandran, The General Position Problem: A Survey, arXiv:2501.19385 [math.CO], 2025. See p. 4.
• Eric Weisstein's World of Mathematics, No-Three-in-a-Line-Problem
• Eric Weisstein's World of Mathematics, Cube
• Index entries for transcendental numbers

Crossrefs

Continued fraction expansion is A132116. - Jonathan Vos Post, Aug 10 2007

Equals twice A093766.

Cf. A343235 (using the reciprocal), A248682.

Programs

Magma

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

Mathematica

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

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

Sage

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

Formula

Equals Integral_{x=0..oo} x^(1/3)/(1+x^2) dx. - Jean-François Alcover, May 24 2013

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

Equals 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

Equals (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)

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

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

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

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

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

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

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

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

Equals 3*Integral_{x = 0..oo} cosh(x)/cosh(3*x) dx. (End)

Equals Integral_{x = 0..oo} log(1+x^3)/x^3 dx. - Amiram Eldar, Aug 20 2020

Equals (27*S - 36)/24, where S = A248682. - Peter Luschny, Jul 22 2022

From Peter Bala, Nov 09 2023: (Start)

For any integer k, Pi/sqrt(3) = Sum_{n >= 0} (1/(n + k + 1/3) - 1/(n - k + 2/3)) = (1/3)*Sum_{n >= 0} (1/(n - k + 1/6) - 1/(n + k + 5/6)).

Equals (3/2)*Sum_{n >= 0} 1/((2*n + 1)*binomial(2*n, n)). (End)