A002388 -id:A002388 - cp's OEIS frontend

A091476 Decimal expansion of Pi^2/4.

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

Offset: 1

Eric W. Weisstein, Jan 13 2004

			2.46740110027233965470862274996903778...

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Ben Hambrecht and Grant Sanderson, The stunning geometry behind this surprising equation, 3Blue1Brown video (2018).
Josef Hofbauer, A simple proof of 1 + 1/2^2 + 1/3^2 + ... = Pi^2/6 and related identities, The American Mathematical Monthly 109:2 (2002), pp. 196-200.
Michael Penn, Neat ways to solve complicated limits., YouTube video, 2023.
T. Piezas III, Golden ratio and nested radicals
Johan Wästlund, Summing inverse squares by euclidean geometry
Eric Weisstein's World of Mathematics, Definite Integral
H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.
Index entries for transcendental numbers

Cf. A002388, A013661, A019669.

evalf(Pi^2/4, 120); # Muniru A Asiru, Sep 18 2018

First[RealDigits[Pi^2/4,10,100]] (* Paolo Xausa, Oct 31 2023 *)

Pi^2/4 \\ Charles R Greathouse IV, Mar 02 2018

2*sumpos(n=1,(2*n-1)^-2) \\ Charles R Greathouse IV, Mar 02 2018

Equals Integral_{x=0..Pi} x*sin(x)/(1+cos(x)^2) dx.
Equals Integral_{x=0..1} log((1+x)/(1-x))/x dx. - Jean-François Alcover, May 13 2013
Equals Integral_{x=0..oo} K_0(x)^2 dx, where K_0 is a modified Bessel function (see Gradstein-Ryshik 6.576.4). - R. J. Mathar, Oct 09 2015
Equals A003881 * A000796. - R. J. Mathar, Oct 09 2015
Equals ... + (-5)^-2 + (-3)^-2 + (-1)^-2 + 1^-2 + 3^-2 + 5^-2 + .... - Charles R Greathouse IV, Mar 02 2018
From A.H.M. Smeets, Sep 18 2018: (Start)
Equals A102753/2.
Equals 2*Sum_{k > 0} 1/(2*k - 1)^2. (End)
Pi^2/4 = Integral_{x = 0..oo} x/sinh(x) dx. More generally, Pi^2/4 = 2*(1 + 1/3^2 + ... + 1/(2*n-1)^2) + Integral_{x = 0..oo} exp(-2*n*x)*x/sinh(x). - Peter Bala, Nov 05 2019
Equals Integral_{x=0..oo} log(x)/(x^2 - 1) dx. - Amiram Eldar, Aug 12 2020
Equals Sum_{n >= 0} 2^(n+1)/((n+1)^2*binomial(2*n+1,n)). See my entry in A002544 dated Apr 18 2017. Cf. A253191. - Peter Bala, Jan 30 2023
From Peter Bala, Nov 16 2023: (Start)
Pi^2/4 = 16*Sum_{k >= 1} k^2/(4*k^2 - 1)^2 = (2*16^2)*Sum_{k >= 1} k^2/((4*k^2 - 1)*(4*k^2 - 9))^2.
The general result, which can be proved using the WZ method (see Wilf for examples of this method), is that for n >= 0 there holds
Pi^2/4 = 16^(n+1)*(2*n + 1)*(2*n)!^4/(4*n)! * Sum_{k >= 1} k^2/( (4*k^2 - 1)*(4*k^2 - 9)*...*(4*k^2 - (2*n+1)^2) )^2. (End)
Equals Re(Polylog(2, 2)). - Mohammed Yaseen, Jul 03 2024
From A.H.M. Smeets, Apr 10 2025: (Start)
Let X(p,q) be the p-th smallest zero of the Laguerre polynomial of order q.
Equals lim_{k -> oo} X(k,k^2).
Equals lim_{q -> oo} X(1,q)*q.
Equals lim_{k -> oo} X(k,k^4)*sqrt(k).
Equals lim_{k -> oo} X(k^3,k^4)/sqrt(k).
More general, let P = log_q(p^2/q), then, for any p, 0 < p <= q, equals lim_{q -> oo} X(p,q)/q^P. (End)
Equals Integral_{x=-1..1} -log(abs(x))/(1 - x^2) dx. - Kritsada Moomuang, May 28 2025

A102753 Decimal expansion of (Pi^2)/2.

Original entry on oeis.org

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

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Feb 10 2005

Also equals the area under the peak-shaped even function f(x)=x/sinh(x).
Proof: For the upper half of the integral, write f(x) = 2x*exp(-x)/(1-exp(-2x)) = sum_{k=1..infinity} 2x*exp(-(2k-1)x) and integrate term by term from zero to infinity. - Stanislav Sykora, Nov 01 2013
Volume of the 4-dimensional unit sphere; the volume of the n-dimensional unit sphere is Pi^(n/2)/gamma(n/2+1) (see n-ball link and A164103). - Rick L. Shepherd, Jun 22 2017
Pi^2/2 is the squared side-length of a square with diagonal Pi. - Wesley Ivan Hurt, Jan 28 2022

			4.9348022005446793094172454999380755676568497036203953132066746881100\ 224112096026215008867018592761159120129568870115720388....

J. Rivaud, Analyse, Séries, Equations différentielles, Mathématiques Supérieures et Spéciales, Premier Cycle Universitaire, Vuibert, 1981, Exercice 2, p. 135.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Middlesex, England: Penguin Books, 1986, p. 53.

G. C. Greubel, Table of n, a(n) for n = 1..10000
T. Amdeberhan, L. Medina and V. H. Moll, The integrals in Gradshteyn and Ryzhik. Part 5: Some trigonometric integrals, equation 2.39, arXiv:0705.2379 [math.CA], 2007.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27
Eric Weisstein's World of Mathematics, Hypersphere.
Eric Weisstein's World of Mathematics, Trigamma Function.
Wikipedia, Hypersphere.
Wikipedia, Volume of an n-ball.
Index entries for transcendental numbers

Cf. A002388, A000796, A019699, A026424, A164103, A164105, A164106, A164108, A248359, A276023.

RealDigits[Pi^2/2, 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2005 *)

PARI
```
Pi^2/2 \\ Michel Marcus, Sep 04 2015
```

Equals psi_1(1/2), where psi_1(x) is the second logarithmic derivative of GAMMA(x).
Equals the volume of revolution of the sine or cosine curve for one half period, Integral_{0,Pi} Sin(x)^2 dx. - Robert G. Wilson v, Dec 15 2005
Equals Sum_{k >=1} 4^k/(k^2*binomial(2*k,k)) [Amdeberhan, Sprugnoli]. - R. J. Mathar, Sep 28 2007
Equals 4*Sum_{k >=1} 1/(2k-1)^2 [Wells].
From  Peter Bala, Nov 05 2019: (Start)
Pi^2/2 = Integral_{x = 0..inf} cosh(x)*x^2/sinh(x)^2 dx.
Pi^2/2 = 5*sum_{k >= 0} binomial(2*k,k)(-1/16)^k*1/(2*k+1)^2.
Pi^2/2 = 10*Integral_{x = 0..1/2}  1/x*log(x + sqrt(1 + x^2)) dx. (End)
Pi^2/20 = 0.1 * Pi^2/2 = Sum_{k>=1} 1/A026424(k)^2. - Amiram Eldar, Aug 17 2020
Conjecture: Pi^2/2 = Sum_{n = -oo..oo} ( cos(Pi*sqrt(n^2+1)) - cos(Pi*n) ) (using the Eisenstein summation convention). - Peter Bala, Oct 08 2021
Pi^2/2 = Integral_{x = -oo..oo} x/sinh(x) dx (see Rivaud reference). - Bernard Schott, Jan 28 2022

A164102 Decimal expansion of 2*Pi^2.

Original entry on oeis.org

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

Offset: 2

R. J. Mathar, Aug 10 2009

Surface area of the 4-dimensional unit sphere. The volume of the 4-dimensional unit sphere is a fourth of this, A102753.

Also decimal expansion of Pi^2/5 = 1.973920..., with offset 1. - Omar E. Pol, Oct 04 2011

			19.739208802178717237668981...

L. A. Santalo, Integral Geometry and Geometric Probability, Addison-Wesley, 1976, see p. 15.

G. C. Greubel, Table of n, a(n) for n = 2..5000
Yann Bernard, Autour des surfaces de Willmore, Images des Mathématiques, CNRS, 2014 (in French).
Fernando C. Marques and André Neves, Min-Max theory and the Willmore conjecture, arXiv:1202.6036 [math.DG], 2012-2013.
H.-J. Seiffert, Problem B-705, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 4 (1991), p. 372; An Application of a Series Expansion for (arcsinx)^2, Solution to Problem B-705, ibid., Vol. 31, No. 1 (1993), pp. 85-86.
Eric Weisstein's World of Mathematics, Hypersphere.
Wikipedia, Hypersphere.
Wikipedia, Willmore conjecture.
Index entries for transcendental numbers.

Cf. A000032, A002388, A002736, A005248, A091476, A013661.

RealDigits[2*Pi^2,10,120][[1]] (* Harvey P. Dale, Apr 19 2012 *)

2*Pi^2 \\ Charles R Greathouse IV, Jan 24 2014

Equals 2*A002388 = 4*A102753.

Pi^2/5 = Sum_{k>=1} Lucas(2*k)/(k^2*binomial(2*k,k)) = Sum_{k>=1} A005248(k)/A002736(k) (Seiffert, 1991). - Amiram Eldar, Jan 17 2022

A195055 Decimal expansion of Pi^2/3.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 04 2011

			3.289868133696452872944830333292050378438...

Marc Briane and Gilles Pagès, Théorie de l'Intégration, Vuibert, 2004, 3ème édition, exercice 12.15, p. 256.

George E. Andrews, Partitions with short sequences and mock theta functions, Proceedings of the National Academy of Sciences, Vol. 102, No. 13 (2005), pp. 4666-4671.
Index entries for transcendental numbers

Cf. A002388, A102753, A091476, 2*A013661, A164102.
Cf. A024916 (partial sums of A000203).
Cf. A001008/A002805, A145426.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*pi^2/3))); // Vincenzo Librandi, Jan 12 2016

RealDigits[Pi^2/3, 10, 105][[1]] (* T. D. Noe, Oct 05 2011 *)

2*zeta(2) \\ Charles R Greathouse IV, Jan 20 2022

sumnumrat(1/x^2,-oo) \\ Charles R Greathouse IV, Jan 20 2022

Equals 3 + A145426.
Equals -Sum_{n>=1} Psi_2(n), where Psi_2 is the tetragamma function. - Istvan Mezo, Oct 25 2012
Equals Integral_{x=0..1} (log(x)/(x - 1))^2 dx. - Jean-François Alcover, Mar 21 2013
Equals Integral_{x=-oo..oo} x^2/sinh(x)^2 dx. - Amiram Eldar, Aug 06 2020
Equals Integral_{x=0..oo} (log(x+1)/x)^2 dx (reference Briane and Pagès). - Bernard Schott, Feb 13 2022
Equals Sum_{n>=1} H(n) * binomial(2*n, n) / (n * 4^n), where H(n) is the n-th harmonic number. - Antonio Graciá Llorente, Apr 04 2025

Extended by T. D. Noe, Oct 05 2011

A212002 Decimal expansion of (2*Pi)^2.

Original entry on oeis.org

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

Offset: 2

Omar E. Pol, Aug 11 2012

This constant appears in Kepler's 3rd Law, T^2 = (2*Pi)^2/GM*a^3 where a is the semi-major axis of a planet orbiting the Sun, T is its period, and GM is the standard gravitational parameter. - Raphie Frank, Dec 13 2012

García & Marco give a generalized zeta regularization by which this is the value of the product of the primes. - Charles R Greathouse IV, Jun 17 2013

			39.4784176043574344753379639995046045412547976289631...

J.-P. Allouche, The zeta-regularized product of odious numbers, arXiv:1906.10532 [math.NT], 2019.
Hyperphysics, Kepler's Laws
E. Muñoz García and R. Pérez Marco, The product over all primes is 4Pi^2, Communications in Mathematical Physics, Vol. 277, No. 1 (2008), pp. 69-81.
Hengguang Li and Jeffrey S. Ovall, A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential, Discrete and Continuous Dynamical Systems Series B, Volume 20, Number 5, July 2015, pp. 1377-1391. Also doi:10.3934/dcdsb.2015.20.1377
Wikipedia, Pendulum.
Index entries for transcendental numbers

Cf. A000796, A002388, A019692, A212003.

RealDigits[(2*Pi)^2,10,120][[1]] (* Harvey P. Dale, Mar 27 2019 *)

4*Pi^2 \\ Charles R Greathouse IV, Jun 17 2013

Equals Product_{k=1..10, gcd(k,10)==1} Gamma(k/10) = Gamma(1/10)*Gamma(3/10)*Gamma(7/10)*Gamma(9/10). - Amiram Eldar, Jun 12 2021

Equals lim_{n->oo} |B(2*n)/B(2*n+2)|*(2*n+1)*(2*n+2), where B(n) denotes the n-th Bernoulli number. - Peter Luschny, Dec 09 2021

A092742 Decimal expansion of 1/Pi^2.

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 12 2004

The asymptotic density of squarefree numbers that are divisible by 5. - Amiram Eldar, Mar 25 2021

			0.101321183642337771443879463209727638904358774672246548845609...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 3.6.1, p. 220.

S. Ramanujan, Some formulas in the analytic theory of numbers Messenger of Mathematics, XLV, 1916, pp. 81-84, Formula (3).
Index entries for transcendental numbers

Cf. A000796 (Pi), A002388 (Pi^2), A091925 (Pi^3), A092425 (Pi^4), A092731 (Pi^5), A092732 (Pi^6), A092735 (Pi^7), A092736 (Pi^8).

Cf. A049541 (1/Pi), A092743 (1/Pi^3), A092744 (1/Pi^4), A092745 (1/Pi^5), A092746 (1/Pi^6), A092747 (1/Pi^7), A092748 (1/Pi^8).

RealDigits[N[1/Pi^2,200]][[1]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *) (* Corrected by Harvey P. Dale, Sep 21 2024. *)
realDigitsRecip[Pi^2] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Sep 21 2024 *)

c=Pi^-2; a=eval(vecextract(Vec(Str(c)),"3..-2"))  \\ M. F. Hasler, Sep 16 2011

A353908 Decimal expansion of Pi^2/36.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, May 10 2022

Ratio between the volume of the stepped pyramid with an infinite number of levels described in A245092 and that of the circumscribed cube (see the first formula).
See also Vaclav Kotesovec's formula (2016) in A175254.
Volume shared by a sphere inscribed in a cube of volume Pi and one of the six pyramids inscribed in the cube. - Omar E. Pol, Sep 01 2024

			0.2741556778080377394120691944410041982031583168677997396225930382283345784...

Index entries for transcendental numbers

Cf. A000203, A000796, A002388, A013661, A019673, A021040, A024916, A072691, A086463, A086729, A091476, A100044, A102753, A175254, A195055, A237271, A237593, A245092.

evalf(Pi^2/36, 121);  # Alois P. Heinz, May 11 2022

RealDigits[Pi^2/36, 10, 100][[1]] (* Amiram Eldar, May 11 2022 *)

PARI
```
Pi^2/36
```
PARI
```
zeta(2)/6
```

Equals lim_{n->oo} A175254(n)/n^3.
Equals A002388/36.
Equals A102753/18.
Equals A195055/12.
Equals A091476/9.
Equals A013661/6.
Equals A100044/4.
Equals A072691/3.
Equals A086463/2.
Equals A086729*2.
Equals A019673^2.
Equals A002388*A021040.
Equals Re(dilog((1+sqrt(3)*i)/2)). - Mohammed Yaseen, Jul 03 2024

A304656 Decimal expansion of Pi*sqrt(3).

Original entry on oeis.org

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

Offset: 1

Peter Luschny, May 16 2018

			5.4413980927026535517822347729264671968521987442782217267096548061643695433790...

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
Wikipedia, Bailey-Borwein-Plouffe formula.
Index entries for transcendental numbers

Cf. A002388, A002391, A093602, A248682.

Maple
```
Pi*sqrt(3): evalf(%, 100);
```

RealDigits[N[StieltjesGamma[0,1/6]-StieltjesGamma[0,5/6],99]][[1]] (* corrected by Harvey P. Dale, Oct 13 2020 *)
RealDigits[Pi Sqrt[3],10,120][[1]] (* Harvey P. Dale, Oct 13 2020 *)

# Use several guard digits when computing.
# BBP formula (9/32) P(1, 64, 6, (16, 8, 0, -2, -1, 0)).
from decimal import Decimal as dec, getcontext
def BBPpisqrt3(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(64)
    for k in range(int(n * 0.5536546824812272) + 1):
        sixk = dec(6 * k)
        s += f * ( dec(16) / (sixk + 1) + dec(8) / (sixk + 2)
                 - dec(2)  / (sixk + 4) - dec(1) / (sixk + 5) )
        f /= g
    return (s * dec(9)) / dec(32)
print(BBPpisqrt3(200))  # Peter Luschny, Nov 03 2023

Equals gamma(0, 1/6) - gamma(0, 5/6) where gamma(n,x) denotes the generalized Stieltjes constants.
Equals PolyGamma[0, 5/6] - PolyGamma[0, 1/6].
Equals 3*sqrt(2*zeta(2)).
Pi^2 = A304656 * A093602.
From Amiram Eldar, Aug 06 2020: (Start)
Equals Sum_{k>=0} 1/((k + 1/3)*(k + 2/3)).
Equals Integral_{x=0..oo} log(1 + 3/x^2) dx. (End)
Equals (27*S - 36)/8, where S = A248682. - Peter Luschny, Jul 22 2022
From Peter Bala, Oct 26 2023: (Start)
sqrt(3)*Pi = 9/2 + 9*Sum_{n >= 1} (-1)^(n+1)/(9*n^2 - 1);
sqrt(3)*Pi = 5 + 10*Sum_{n >= 1} 1/((4*n^2 - 1)*(9*n^2 - 1)) = 43/8 + 8*Sum_{n >= 2} (-1)^n/((n^2 - 1)*(9*n^2 - 1));
sqrt(3)*Pi = 1765/324 - (80/9)*Sum_{n >= 2} 1/((n^2 - 1)*(4*n^2 - 1)*(9*n^2 - 1)).
The following two series representations for the constant
sqrt(3)*Pi = 72 * Sum_{n >= 0} (2*n + 1)/((6*n + 1)*(6*n + 3)*(6*n + 5)) and
sqrt(3)*Pi = 8192/1485 - 860160 * Sum_{n >= 0} (2*n + 3)/((6*n + 1)*(6*n + 3)*...*(6*n + 17)) appear to generalize as follows:
for k >= 0, sqrt(3)*Pi = c(k) + (-1)^k*d(k)*Sum_{n >= 0} (2*n + 2*k + 1)/((6*n + 1)*(6*n + 3)*...*(6*n + 12*k + 5)), where c(k) is a rational number approximating sqrt(3)*Pi and d(k) = (6*k + 1)! * 2^(6*k+3) / 3^(3*k-2).
The first few values of c(k) for k >= 0 are [0, 8192/1485, 11341398016/2085060285, 62809601736704/11542783997745, 889063287831973723111424/ 163388820474305231710905, ...].
The following two series representations for the constant
sqrt(3)*Pi = 256/45 - 2560*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 3)*...*(6*n + 11)) and
sqrt(3)*Pi = 337117184/62026965 + 2018508800*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 3)*...*(6*n + 23)) appear to generalize as follows:
for k >= 0, sqrt(3)*Pi = c(k) - (-1)^k*d(k)*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 3)*...*(6*n + 12*k + 11)), where c(k) is a rational number approximating sqrt(3)*Pi and d(k) = (6*k + 5)! * 2^(6*k+6) / 3^(3*k+1).
The first few values of c(k) for k >= 0 are [256/45, 337117184/62026965, 1732370763874304/318357429615225, 733187044080753836032/134742553582636674675, 6361250411469779336874164224/1169047010493653932891525275, ...]. (End)
For arbitrary integer k, Pi*sqrt(3) = Sum_{n >= 0} (1/(n - k + 1/6) - 1/(n + k + 5/6)) = Sum_{n >= 0} (1/(n + k + 7/6) - 1/(n - k - 1/6)). - Peter Bala, Jul 10 2024

A092736 Decimal expansion of Pi^8.

Original entry on oeis.org

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

Offset: 4

Mohammad K. Azarian, Apr 12 2004

			9488.53101607057400712857550390676579669717947164108269211009141506690890...

G. C. Greubel, Table of n, a(n) for n = 4..10000
Index entries for transcendental numbers

Cf. A000796, A002388, A092425.

R:= RealField(100); (Pi(R))^8; // G. C. Greubel, Mar 09 2018

evalf(Pi^8,100); # Wesley Ivan Hurt, Jan 28 2017

RealDigits[Pi^8, 10, 100][[1]] (* G. C. Greubel, Mar 09 2018 *)

Pi^8 \\ Charles R Greathouse IV, Jan 29 2017

A118858 Decimal expansion of log(2)/Pi^2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, May 02 2006

			0.07023049277268287640...

Derrick Norman Lehmer, Asymptotic Evaluation of Certain Totient Sums, American Journal of Mathematics, Vol. 22, No. 4 (1900), pp. 293-335.
Eric Weisstein's World of Mathematics, Primitive Pythagorean Triple.

Cf. A249750, A328499, A002162, A002388, A284983.

RealDigits[Log[2]/Pi^2, 10, 100][[1]] (* Amiram Eldar, Jun 25 2021 *)

log(2)/Pi^2 \\ Michel Marcus, Jun 25 2021

Equals lim_{n->oo} A328499(n)/n. - Amiram Eldar, Jun 25 2021

Equals Integral_{z>=0} z*sech(Pi*z)^2. - Peter Luschny, Aug 03 2021

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A212002 Decimal expansion of (2*Pi)^2.

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A092742 Decimal expansion of 1/Pi^2.

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A353908 Decimal expansion of Pi^2/36.