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 A002388 Decimal expansion of Pi^2. (Formerly M4596 N1961) 45
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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 REFERENCES 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). LINKS 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? Simon Plouffe, Pi^2 to 10000 digits Simon Plouffe, Plouffe's Inverter, Pi^2 to 10000 digits FORMULA 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 * int {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 EXAMPLE 9.869604401089358618834490999876151135313699407240790626413349376220044... MAPLE Digits:=100: evalf(Pi^2); # Wesley Ivan Hurt, Jul 13 2014 MATHEMATICA RealDigits[Pi^2, 10, 111][[1]] (* Robert G. Wilson v *) PROG (PARI) { 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 (MAGMA) R:= RealField(100); Pi(R)^2; // G. C. Greubel, Mar 08 2018 CROSSREFS Cf. A102753, A058284, A019670, A091925, A093954, A093602, A304656. Sequence in context: A086053 A129269 A094145 * A248080 A278828 A011116 Adjacent sequences:  A002385 A002386 A002387 * A002389 A002390 A002391 KEYWORD nonn,cons AUTHOR EXTENSIONS More terms from Robert G. Wilson v, Dec 15 2005 STATUS approved

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Last modified December 16 12:31 EST 2018. Contains 318160 sequences. (Running on oeis4.)