OFFSET
1,2
COMMENTS
Value of the Borwein-Borwein function I_3(a,b) for a = b = 1. - Stanislav Sykora, Apr 16 2015
The area of a circle circumscribing a unit-area regular hexagon. - Amiram Eldar, Nov 05 2020
REFERENCES
George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), pp. 120-121.
L. B. W. Jolley, Summation of Series, Dover (1961), No. 261, pp. 48, 49, (and No. 275).
LINKS
Xavier Gourdon and Pascal Sebah, Collection of series for Pi (see paragraph 7).
Su Hu and Min-Soo Kim, A generalization of Wallis' formula, arXiv:2201.09674 [math.NT], 2022.
Richard Kershner, The Number of Circles Covering a Set, American Journal of Mathematics, 61(3), 665-671.
MIT Integration Bee, 2023 MIT Integration Bee - Finals, Problem 1.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), C6.2.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See (9) p. 146.
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean, equations 26-32.
FORMULA
Equals 2*sqrt(3)*Pi/9 = 1 + 1/6 + 1/30 + 1/140 + 1/630 + 1/2772 + 1/12012 + ...
Equals m*I_3(m,m) = m*Integral_{x>=0} (x/(m^3+x^3)), for any m>0. - Stanislav Sykora, Apr 16 2015
Equals Integral_{x>=0} (1/(1+x^3)) dx. - Robert FERREOL, Dec 23 2016
From Peter Bala, Oct 27 2019: (Start)
Equals 3/4*Sum_{n >= 0} (n+1)!*(n+2)!/(2*n+3)!.
Equals Sum_{n >= 1} 3^(n-1)/(n*binomial(2*n,n)).
Equals 2*Sum_{n >= 1} 1/(n*binomial(2*n,n)). See Boros and Moll, pp. 120-121.
Equals Integral_{x = 0..1} 1/(1 - x^3)^(1/3) dx = Sum_{n >= 0} (-1)^n*binomial(-1/3,n) /(3*n + 1).
Equals 2*Sum_{n >= 1} 1/((3*n-1)*(3*n-2)) = 2*(1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ...) (added Oct 30 2019). (End)
Equals Product_{k>=1} 9*k^2/(9*k^2 - 1). - Amiram Eldar, Aug 04 2020
From Peter Bala, Dec 13 2021: (Start)
Equals (2/3)*A093602.
Conjecture: for k >= 0, 2*sqrt(3)*Pi/9 = (3/2)^k * k!*Sum_{n = -oo..oo} (-1)^n/ Product_{j = 0..k} (3*n + 3*j + 1). (End)
Equals (3/4)*S - 1, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Integral_{x=0..Pi/2} tan(x)^(1/3)/(sin(2*x) + 1) dx. See MIT Link. - Joost de Winter, Aug 26 2023
Continued fraction: 1/(1 - 1/(7 - 12/(12 - 30/(17 - ... - 2*n*(2*n - 1)/((5*n + 2) - ... ))))). See A000407. - Peter Bala, Feb 20 2024
Equals Sum_{n>=2} 1/binomial(n, floor(n/2)); and trivially if "floor" is replaced by "ceiling". - Richard R. Forberg, Aug 30 2024
Equals Product_{k>=2} (1 + (-1)^k/A001651(k)). - Amiram Eldar, Nov 22 2024
EXAMPLE
1.2091995761561452337293855050947704881893774987284937170465899569254...
MATHEMATICA
RealDigits[2 Sqrt[3] Pi/9, 10, 100][[1]]
PROG
(PARI) a = 2*Pi/(3*sqrt(3)) \\ Stanislav Sykora, Apr 16 2015
CROSSREFS
Cf. A091682 (Sum_{i >= 0} (i!)^2/(2*i)!).
KEYWORD
AUTHOR
Bruno Berselli, Mar 06 2015
STATUS
approved