A091682 -id:A091682 - cp's OEIS frontend

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

Bruno Berselli, Mar 06 2015

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

			1.2091995761561452337293855050947704881893774987284937170465899569254...

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).

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

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)!).

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

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

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

A073016 Decimal expansion of Sum_{n>=1} 1/binomial(2n,n).

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 03 2002

			0.7363998587187150779097951683649234960631258329094979056821966523...

Jean-Marie Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.1.q' pp. 247 and 439.

Simon Plouffe, sum(1/binomial(2n,n), n=1..infinity)
Renzo Sprugnoli, Sums of Reciprocals of the Central Binomial Coefficients, INTEGERS, 6 (2006), #A27, page 9.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient

Cf. A000984 (central binomial coefficients), A091682, A248179.

RealDigits[ N[ (9 + 2*Sqrt[3]*Pi)/27, 110]] [[1]]

(2*Pi*sqrt(3)+9)/27 \\ Michel Marcus, Aug 10 2014

Equals (9 + 2*sqrt(3)*Pi)/27.
Equals A091682 - 1.
Equals Integral_{x=0..Pi/2} cos(x)/(2 - cos(x))^2 dx. - Amiram Eldar, Aug 19 2020
From Bernard Schott, May 12 2022: (Start)
Equals Sum_{n>=1} (n!)^2 / (2*n)!.
Equals A248179 / 2. (End)

A248179 Decimal expansion of (2/27)(9 + 2sqrt(3)*Pi).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 03 2014

			1.472799717437430155819590336729846992126251665818995811364393304616943...

J.-M. Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.1.q', p. 247.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. A073016, A091682, A248180.

r = 2/27 (9 + 2 Sqrt[3] \[Pi]); u = RealDigits[N[r, 200]][[1]]

2*(9+sqrt(12)*Pi)/27 \\ Charles R Greathouse IV, Sep 28 2022

Equals Sum_{h >= 0} 1/binomial(2*h+1, h).
From Amiram Eldar, Nov 16 2021: (Start)
Equals 1 + Integral_{x>=0} 1/(x^2 + x + 1)^2 dx.
Equals 1 + Integral_{x>=1} 1/(x^2 - x + 1)^2 dx.
Equals Integral_{x=0..1} 1/(x^2 - x + 1)^2 dx. (End)
From Bernard Schott, Mar 18 2022: (Start)
Equals 2 * Sum_{n >= 1} (n!)^2/(2*n)!.
Equals 2 * A073016.
Equals hypergeometric function 2F1([1, 2], [3/2], x) at x=1/4. (End)

A307086 Decimal expansion of 4(5 - sqrt(5)log(phi))/25, where phi is the golden ratio (A001622).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 23 2019

Decimal expansion of the alternating sum of the reciprocals of the central binomial coefficients (A000984).

			1/1 - 1/2 + 1/6 - 1/20 + 1/70 - 1/252 + ... = 0.62783642361439838444422670681975782983017172698388...

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
Eric Weisstein's World of Mathematics, Central Binomial Coefficient
Index entries for transcendental numbers

Cf. A000984, A001622, A073016, A086465, A091682, A145433, A145434.

RealDigits[4 (5 - Sqrt[5] Log[GoldenRatio])/25, 10, 120][[1]]

4*(5 - sqrt(5)*log((sqrt(5)+1)/2))/25 \\ Charles R Greathouse IV, May 15 2019

Equals Sum_{k>=0} (-1)^k/binomial(2*k,k).

Equals Sum_{k>=0} (-1)^k*(k!)^2/(2*k)!.

cp's OEIS Frontend

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

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A073016 Decimal expansion of Sum_{n>=1} 1/binomial(2n,n).

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A248179 Decimal expansion of (2/27)*(9 + 2*sqrt(3)*Pi).

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A307086 Decimal expansion of 4*(5 - sqrt(5)*log(phi))/25, where phi is the golden ratio (A001622).

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A248179 Decimal expansion of (2/27)(9 + 2sqrt(3)*Pi).

A307086 Decimal expansion of 4(5 - sqrt(5)log(phi))/25, where phi is the golden ratio (A001622).