A225016 Decimal expansion of Pi^3/8.
3, 8, 7, 5, 7, 8, 4, 5, 8, 5, 0, 3, 7, 4, 7, 7, 5, 2, 1, 9, 3, 4, 5, 3, 9, 3, 8, 3, 3, 8, 7, 6, 7, 4, 4, 0, 0, 2, 7, 8, 1, 6, 1, 0, 7, 0, 7, 3, 5, 6, 3, 8, 4, 6, 1, 7, 6, 8, 0, 6, 7, 2, 6, 2, 9, 7, 5, 7, 9, 9, 3, 6, 4, 6, 8, 3, 2, 1, 3, 2, 5, 4, 6, 9, 5, 8, 3, 7, 6, 2, 9, 0, 7, 5, 3, 6, 0, 7, 7, 4
Offset: 1
Examples
3.875784585037477521934539383387674400278161070735638461768067262975799364683...
Programs
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Mathematica
RealDigits[Pi^3/8, 10, 100][[1]]
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PARI
Pi^3/8 \\ Charles R Greathouse IV, Oct 01 2022
Formula
Equals Integral_{x>0} log(x)^2/(1+x^2) dx.
Equals Integral_{x=0..Pi/2} log(tan(x))^2 dx.
Equals Integral_{x=0..Pi/2} log(sin(x)^3)*log(sin(x))-(3*Pi/2)*log(2)^2 dx.
Equals (27/7) * Sum_{k>=0} binomial(2*k, k)/((2*k+1)^3*16^k);
Equals (27/7) * 4F3([1/2, 1/2, 1/2, 1/2], [3/2, 3/2, 3/2], 1/4), where pFq() is the generalized hypergeometric function.
From Amiram Eldar, Aug 21 2020: (Start)
Equals Integral_{x=0..oo} x^2/cosh(x) dx.
Equals 2 + Integral_{x=0..oo} x^2 * exp(-x) * tanh(x) dx. (End)
From Gleb Koloskov, Jun 15 2021: (Start)
Equals 2*Integral_{x=0..1} log(x)^2/(1+x^2) dx.
Equals 2*Integral_{x=1..oo} log(x)^2/(1+x^2) dx.
Equals 2*(-1)^n*Integral_{x=-1/e..0} W(n,x)*(1-W(n,x))*log(-W(n,x))^2/x/(1-W(n,x)^4) dx, where W=LambertW, for n=0 and n=-1. (End)
Extensions
Offset corrected by Rick L. Shepherd, Jan 01 2014