A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.

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

Offset: 1

Author

Bruno Berselli, Mar 06 2015

Keywords

• nonn
• cons
• easy

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

Examples

			1.2091995761561452337293855050947704881893774987284937170465899569254...
		

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.
• Index entries for transcendental numbers

Crossrefs

Cf. A000796, A001651, A002194, A093602, A248181, A257096, A257097, A186706.

Cf. A010815, A073010, A086089, A214549, A256923, A275486.

Cf. A091682 (Sum_{i >= 0} (i!)^2/(2*i)!).

Programs

Mathematica

RealDigits[2 Sqrt[3] Pi/9, 10, 100][[1]]

PARI

a = 2*Pi/(3*sqrt(3)) \\ Stanislav Sykora, Apr 16 2015

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

Equals 2*A073010 = 1/A086089 = sqrt(A214549) = exp(A256923) = A275486/2. - Hugo Pfoertner, Nov 22 2024

Equals 1 - (1/6) * Sum_{n>=1} A010815(n)/n. - Friedjof Tellkamp, Apr 05 2025

Equals A248181 - 2. - Pontus von Brömssen, Apr 05 2025