The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A093602 Decimal expansion of Pi/sqrt(3) = sqrt(2*zeta(2)). 23
 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 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 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. Eric Weisstein's World of Mathematics, No-Three-in-a-Line-Problem Eric Weisstein's World of Mathematics, Cube Index entries for transcendental numbers 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) 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 Continued fraction expansion is A132116. - Jonathan Vos Post, Aug 10 2007 Equals twice A093766. Cf. A343235 (using the reciprocal), A248682. Sequence in context: A092515 A193032 A127454 * A011469 A140457 A358286 Adjacent sequences: A093599 A093600 A093601 * A093603 A093604 A093605 KEYWORD easy,nonn,cons,changed AUTHOR Lekraj Beedassy, May 14 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 3 10:02 EST 2023. Contains 367539 sequences. (Running on oeis4.)