A102753 -id:A102753 - cp's OEIS frontend

A002388 Decimal expansion of Pi^2.

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

Offset: 1

N. J. A. Sloane

Also equals the volume of revolution of the sine or cosine curve for one full period, Integral_{x=0..2*Pi} sin(x)^2 dx. - Robert G. Wilson v, Dec 15 2005

Equals Sum_{n>0} 20/A026424(n)^2 where A026424 are the integers such that the number of prime divisors (counted with multiplicity) is odd. - Michel Lagneau, Oct 23 2015

			9.869604401089358618834490999876151135313699407240790626413349376220044...

W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 76.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary (The Treatise on the Circumference), Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.
D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices AMS, 52 (No. 5 2005), 502-514.
David H. Bailey, Jonathan M. Borwein, Andrew Mattingly, and Glenn Wightwick, The Computation of Previously Inaccessible Digits of (Pi)^2 and Catalan's Constant, Notices AMS, 60 (No. 7 2013), 844-854.
N. D. Elkies, Why is (Pi)^2 so close to 10?
Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
Simon Plouffe, Pi^2 to 10000 digits.
Simon Plouffe, Plouffe's Inverter, Pi^2 to 10000 digits.
Wikipedia, Bailey-Borwein-Plouffe formula.
Herbert S. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science, Vol 3, No 4 (1999).
Index entries for sequences related to the number Pi.
Index entries for transcendental numbers.

Cf. A102753, A058284, A019670, A091925, A093954, A093602, A304656.

R:= RealField(100); Pi(R)^2; // G. C. Greubel, Mar 08 2018

Digits:=100: evalf(Pi^2); # Wesley Ivan Hurt, Jul 13 2014

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

default(realprecision, 20080); x=Pi^2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002388.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

# Use some guard digits when computing.
# BBP formula (9 / 8) P(2, 64, 6, (16, -24, -8, -6,  1, 0)).
from decimal import Decimal as dec, getcontext
def BBPpi2(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) ** 2 - dec(24) / (sixk + 2) ** 2
                 - dec(8)  / (sixk + 3) ** 2 - dec(6)  / (sixk + 4) ** 2
                 + dec(1)  / (sixk + 5) ** 2 )
        f /= g
    return (s * dec(9)) / dec(8)
print(BBPpi2(200))  # Peter Luschny, Nov 03 2023

Pi^2 = 11/2 + 16 * Sum_{k>=2} (1+k-k^3)/(1-k^2)^3. - Alexander R. Povolotsky, May 04 2009
Pi^2 = 3*(Sum_{n>=1} ((2*n+1)^2/Sum_{k=1..n} k^3)/4 - 1). - Alexander R. Povolotsky, Jan 14 2011
Pi^2 = (3/2)*(Sum_{n>=1} ((7*n^2+2*n-2)/(2*n^2-1)/(n+1)^5) - zeta(3) - 3*zeta(5) + 22 - 7*polygamma(0,1-1/sqrt(2)) + 5*sqrt(2)*polygamma(0,1-1/sqrt(2)) - 7*polygamma(0,1+1/sqrt(2)) - 5*sqrt(2)*polygamma(0,1+1/sqrt(2)) - 14*EulerGamma). - Alexander R. Povolotsky, Aug 13 2011
Also equals 32*Integral_{x=0..1} arctan(x)/(1+x^2) dx. - Jean-François Alcover, Mar 25 2013
From Peter Bala, Feb 05 2015: (Start)
Pi^2 = 20 * Integral_{x = 0 .. log(phi)} x*coth(x) dx, where phi = (1/2)*(1 + sqrt(5)) is the golden ratio.
Pi^2 = 10 * Sum_{k >= 0} binomial(2*k,k)*(1/(2*k + 1)^2)*(-1/16)^k. Similar series expansions hold for Pi/3 (see A019670) and (7*/216)*Pi^3 (see A091925).
The integer sequences A(n) := 2^n*(2*n + 1)!^2/n! and B(n) := A(n)*( Sum_{k = 0..n} binomial(2*k,k)*1/(2*k + 1)^2*(-1/16)^k ) both satisfy the second order recurrence equation u(n) = (24*n^3 + 44*n^2 + 2*n + 1)*u(n-1) + 8*(n - 1)*(2*n - 1)^5*u(n-2). From this observation we can obtain the continued fraction expansion Pi^2/10 = 1 - 1/(72 + 8*3^5/(373 + 8*2*5^5/(1051 + ... + 8*(n - 1)*(2*n - 1)^5/((24*n^3 + 44*n^2 + 2*n + 1) + ... )))). Cf. A093954. (End)
Pi^2 = A304656 * A093602 = (gamma(0, 1/6) - gamma(0, 5/6))*(gamma(0, 2/6) - gamma(0, 4/6)), where gamma(n,x) are the generalized Stieltjes constants. This formula can also be expressed by the polygamma function. - Peter Luschny, May 16 2018
Equals 8 + Sum_{k>=1} 1/(k^2 - 1/4)^2 = -8 + Sum_{k>=0} 1/(k^2 - 1/4)^2. - Amiram Eldar, Aug 21 2020
From Peter Bala, Dec 10 2021: (Start)
Pi^2 = (2^6)*Sum_{n >= 1} n^2/(4*n^2 - 1)^2 = (2^11)*Sum_{n >= 1} n^2/ ((4*n^2 - 1)^2*(4*n^2 - 3^2)^2) = ((2^19)*(3^2)/7) * Sum_{n >= 1} n^2/((4*n^2 - 1)^2*(4*n^2 - 3^2)^2*(4*n^2 - 5^2)^2).
More generally, it appears that for k >= 0 we have Pi^2 = (2*k+1)*2^(4*k+6) * (2*k)!^4/(4*k)! * Sum_{n >= 1} n^2/((4*n^2 - 1)^2*...*(4*n^2 - (2*k+1)^2)^2).
It also appears that for k >= 0 we have Pi^2 = (-1)^k * 2^(6*k+8)*(2*k+1)^3/(6*k+1) * ((2*k)!^6 * (3*k)!)/(k!^3 * (6*k)!) * Sum_{n >= 1} n^2/((4*n^2 - 1)^3*...*(4*n^2 - (2*k+1)^2)^3). (End)
From Peter Bala, Oct 27 2023: (Start)
Pi^2 = 10 - Sum_{n >= 1} 1/(n*(n + 1))^3.
Pi^2 = 6217/630 + (648/35)*Sum_{n >= 1} 1/(n*(n + 1)*(n + 2)*(n + 3))^3.
The general result (verified using the WZ method - see Wilf) is : for n >= 0,
Pi^2 = A(n) + (-1)^(n+1) * B(n)*Sum_{k >= 1} 1/(k*(k + 1)*...*(k + 2*n + 1))^3, where A(n) = 10 - Sum_{i = 1..n} (-1)^(i+1) * (56*i^2 + 24*i + 3)*(2*i)!^3*(3*i)!/(2*i^2*(2*i + 1)*(6*i + 1)!*i!^3) and B(n) = (2*n + 1)!^6 * (3*n)! / ( (2*n + 1)*(6*n + 1)!*n!^3 ).
Letting n -> oo gives the fast converging alternating series
Pi^2 = 10 - Sum_{i >= 1} (-1)^(i+1) * (56*i^2 + 24*i + 3)*(2*i)!^3 * (3*i)!/(2*i^2*(2*i + 1)*(6*i + 1)!*i!^3). The i-th summand of the series is asymptotic to (14/3) * 1/(i^2 * 27^i) so taking 70 terms of the series gives a value for Pi^2 accurate to more than 100 decimal places.
The series representation Pi^2 = 3*Sum_{k >= 1} (2*k)/k^3 can be accelerated to give the faster converging series
Pi^2 = 99/10 - (8/5)*Sum_{k >= 1} (2*k + 2)/(k*(k + 1)*(k + 2))^3 and
Pi^2 = 54715/5544 + (41472/385)*Sum_{k >= 1} (2*k + 4)/(k*(k + 1)*(k + 2)*(k + 3)*(k + 4))^3.
The general result is: for n >= 1, Pi^2 = C(n) + (-1)^n * D(n)*Sum_{k >= 1} (2*k + 2*n)/(k*(k + 1)*...*(k + 2*n))^3, where C(n) = A(n) - 10*(-1)^n*(3*n)!*(2*n)!^3/((2*n + 1)*n!^3*(6*n + 1)!) and D(n) = (2*n)!^6 * (3*n)! / ( 2*n*(6*n - 1)!*n!^3 ). (End)
Equals 9 + 3*Sum_{n>=1} 1/((n^2*(n+1)^2)). - Davide Rotondo, May 29 2025

More terms from Robert G. Wilson v, Dec 15 2005

A111003 Decimal expansion of Pi^2/8.

Original entry on oeis.org

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

Offset: 1

Sam Alexander, Oct 01 2005

According to Beckmann, Euler discovered the formula for this number as a sum of squares of reciprocals of odd numbers, along with similar formulas for Pi^2/6 and Pi^2/12. - Alonso del Arte, Apr 01 2013

Equals the asymptotic mean of the abundancy index of the odd numbers. - Amiram Eldar, May 12 2023

			1.23370055013616982735431137498451889191421242590509882830166867202...
1 + 1/9 + 1/25 + 1/49 + 1/81 + 1/121 + 1/169 + 1/225 + ... - _Bruno Berselli_, Mar 06 2017

F. Aubonnet, D. Guinin and B. Joppin, Précis de Mathématiques, Analyse 2, Classes Préparatoires, Premier Cycle Universitaire, Bréal, 1990, Exercice 908, pages 82 and 91-92.
Petr Beckmann, A History of Pi, 5th Ed. Boulder, Colorado: The Golem Press (1982): p. 153.
George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 122.
Calvin C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 98.
L. B. W. Jolley, Summation of Series, Dover (1961).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 54.

Eric Weisstein's World of Mathematics, Dirichlet Lambda Function. See (5).
Index entries for transcendental numbers

Cf. A013661 (Pi^2/6), A002388, A102753, A091476, A309891.

RealDigits[Pi^2/8, 10, 105][[1]] (* Robert G. Wilson v *)

Pi^2/8 \\ Charles R Greathouse IV, Dec 04 2016

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

Equals 1 + 1/(2*3) + (1/3)*(1*2)/(3*5) + (1/4)*(1*2*3)/(3*5*7) + ... [Jolley eq 276]
Equals Sum_{k >= 1} 1/(2*k - 1)^2 [Clawson and Wells]. - Alonso del Arte, Aug 15 2012
Equals 2*(Integral_{t=0..1} sqrt(1 - t^2) dt)^2. - Alonso del Arte, Mar 29 2013
Equals Sum_{k >= 1} 2^k/(k^2*binomial(2*k, k)). - Jean-François Alcover, Apr 29 2013
Equals Integral_{x=0..1} log((1+x^2)/(1-x^2))/x dx. - Bruno Berselli, May 13 2013
Equals limit_{p->0} Integral_{x=0..Pi/2} x*tan(x)^p dx. [Jean-François Alcover, May 17 2013, after Boros & Moll p. 230]
Equals A002388/8 = A102753/4 = A091476/2. - R. J. Mathar, Oct 13 2015
Equals Integral_{x>=0} x*K_0(x)*K_1(x)dx where K are modified Bessel functions [Gradsteyn-Ryzhik 6.576.4]. - R. J. Mathar, Oct 22 2015
Equals (3/4)*zeta(2) = (3/4)*A013661. - Wolfdieter Lang, Sep 02 2019
From Amiram Eldar, Jul 17 2020: (Start)
Equals -Integral_{x=0..1} log(x)/(1 - x^2) dx = Integral_{x>=1} log(x)/(x^2-1) dx.
Equals -Integral_{x=0..oo} log(x)/(1 - x^4) dx.
Equals Integral_{x=0..oo} arctan(x)/(1 + x^2) dx. (End)
Equals Integral_{x=0..1} log(1+x+x^2+x^3)/x dx (Aubonnet). - Bernard Schott, Feb 04 2022
Equals Sum_{n>=1} A309891(n)/n^2. - Friedjof Tellkamp, Jan 25 2025
Equals lambda(2), where lambda is the Dirichlet lambda function. - Michel Marcus, Aug 15 2025

More terms from Robert G. Wilson v, Oct 04 2005

A091476 Decimal expansion of Pi^2/4.

Original entry on oeis.org

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

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

A075561 Domination number for kings' graph K(n).

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 9, 9, 9, 16, 16, 16, 25, 25, 25, 36, 36, 36, 49, 49, 49, 64, 64, 64, 81, 81, 81, 100, 100, 100, 121, 121, 121, 144, 144, 144, 169, 169, 169, 196, 196, 196, 225, 225, 225, 256, 256, 256, 289, 289, 289, 324, 324, 324, 361, 361, 361, 400, 400

Offset: 1

N. J. A. Sloane, Oct 16 2002

Also the lower independence number of the n X n knight graph. - Eric W. Weisstein, Aug 01 2023

John J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, 2004, p. 102.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Irene Choi, Shreyas Ekanathan, Aidan Gao, Tanya Khovanova, Sylvia Zia Lee, Rajarshi Mandal, Vaibhav Rastogi, Daniel Sheffield, Michael Yang, Angela Zhao, and Corey Zhao, The Struggles of Chessland, arXiv:2212.01468 [math.HO], 2022.
Matthew D. Kearse and Peter B. Gibbons, Computational Methods and New Results for Chessboard Problems, Centre for Discrete Mathematics and Theoretical Computer Science, CDMTCS-133, May 2000.
Matthew D. Kearse and Peter B. Gibbons, Computational Methods and New Results for Chessboard Problems, Australasian Journal of Combinatorics 23 (2001), 253-284.
Stephan Mertens, Domination Polynomials of the Grid, the Cylinder, the Torus, and the King Graph, arXiv:2408.08053 [math.CO], 2024. See p. 15.
Eric Weisstein's World of Mathematics, Domination Number
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Kings Problem
Eric Weisstein's World of Mathematics, Lower Independence Number
Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A189889, A075458, A006075, A102753.

Table[Floor[(n + 2)/3]^2, {n, 50}] (* Vaclav Kotesovec, May 13 2012 *)
LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 1, 1, 4, 4, 4, 9}, 20] (* Eric W. Weisstein, Jun 20 2017 *)
CoefficientList[Series[(-1 - x^3)/((-1 + x)^3 (1 + x + x^2)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 20 2017 *)

Vec(-x*(x+1)*(x^2-x+1)/((x-1)^3*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Oct 06 2014

a(n) = floor((n+2)/3)^2. - Vaclav Kotesovec, May 13 2012
G.f.: -x*(x+1)*(x^2-x+1) / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, Oct 06 2014
E.g.f.: exp(-x/2)*(exp(3*x/2)*(5 + 3*x*(3 + x)) + (6*x - 5)*cos(sqrt(3)*x/2) + sqrt(3)*(3 + 2*x)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Oct 17 2022
Sum_{n>=1} 1/a(n) = Pi^2/2 (A102753). - Amiram Eldar, Nov 03 2022

More terms added from Vaclav Kotesovec, May 13 2012

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

A164106 Decimal expansion of 16*Pi^3/105.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 7-dimensional unit sphere.

			Equals 4.72476597033140116959639086736783164986290111...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Hypersphere, Wolfram Mathworld.
Wikipedia, Hypersphere
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164105, A164108.

RealDigits[16*Pi^3/105, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

16*Pi^3/105 \\ G. C. Greubel, Apr 09 2017

Equals 16*A091925/105 = A164107/7 .

A164108 Decimal expansion of Pi^4/24.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 10 2009

Volume of the 8-dimensional unit sphere.

			4.0587121264167682181850138620293796354053160696952259038...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Cornel Ioan Vălean, Problem 11993, The American Mathematical Monthly, Vol. 124, No. 7 (2017), p. 659; An Integral Related to Euler Sums, Solution to Problem 11993 by Abdelhak Berkane, ibid., Vol. 126, No. 4 (2019), pp. 373-374.
Eric Weisstein's World of Mathematics, Hypersphere.
Wikipedia, n-sphere.
Index entries for transcendental numbers

Cf. A000796, A019699, A102753, A164103, A164105, A164106.

RealDigits[Pi^4/24, 10, 100][[1]] (* G. C. Greubel, Apr 09 2017 *)

PARI
```
Pi^4/24 \\ G. C. Greubel, Apr 09 2017
```

Equals A164109/8 = A092425/24 = A072691*A102753.

Pi^4/240 = -Integral_{x=0..1} log(1-x)*log(1+x)^2/x dx (Vălean, 2017). - Amiram Eldar, Mar 26 2022

cp's OEIS Frontend

A002388 Decimal expansion of Pi^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A111003 Decimal expansion of Pi^2/8.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A091476 Decimal expansion of Pi^2/4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A164102 Decimal expansion of 2*Pi^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A195055 Decimal expansion of Pi^2/3.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A075561 Domination number for kings' graph K(n).

Views

Author

Keywords

Comments

References