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A002388 Decimal expansion of Pi^2.
(Formerly M4596 N1961)
38
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

Bailey, David H., Jonathan M. Borwein, Andrew Mattingly, and Glenn Wightwick. "The Computation of Previously Inaccessible Digits of π^2 and Catalan’s Constant," Notices AMS, 60 (No. 7 2013), 844-854.

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

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

Index entries for sequences related to the number Pi

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)

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

CROSSREFS

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

Sequence in context: A086053 A129269 A094145 * A248080 A011116 A106334

Adjacent sequences:  A002385 A002386 A002387 * A002389 A002390 A002391

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified September 25 04:48 EDT 2016. Contains 276525 sequences.