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A244978
Decimal expansion of Pi/32.
7
0, 9, 8, 1, 7, 4, 7, 7, 0, 4, 2, 4, 6, 8, 1, 0, 3, 8, 7, 0, 1, 9, 5, 7, 6, 0, 5, 7, 2, 7, 4, 8, 4, 4, 6, 5, 1, 3, 1, 1, 6, 1, 5, 4, 3, 7, 3, 0, 4, 7, 2, 0, 5, 6, 9, 0, 5, 4, 6, 7, 0, 1, 8, 5, 0, 9, 6, 1, 9, 2, 6, 2, 6, 9, 6, 4, 4, 4, 0, 3, 1, 2, 0, 7, 1, 2, 6, 0, 8, 8, 2, 9, 1, 9, 4, 1, 1, 5, 8, 3, 7, 4, 4, 4, 2, 1
OFFSET
0,2
REFERENCES
George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), Chapter 13 A Master Formula, p. 250.
LINKS
FORMULA
Equals Integral_{x = 0..1} x^2/(1 + x^2)^3 dx.
Also equals beta(3/2, 1/2)/16, where 'beta' is Euler's beta function.
From Peter Bala, Oct 27 2019: (Start)
Equals Integral_{x = 0..1} x^4*sqrt(1 - x^2) dx = Integral_{x = 0..1} x^5*sqrt(1 - x^4) dx = Integral_{x = 0..1} x^7*sqrt(1 - x^16) dx.
Equals Integral_{x >= 0} x^4/(1 + x^2)^4 dx. (End)
From Amiram Eldar, Jul 13 2020: (Start)
Equals Integral_{x=0..oo} dx/(x^2 + 4)^2.
Equals Sum_{k>=1} sin(k)^3*cos(k)^3/k. (End)
From Peter Bala, Dec 08 2021: (Start)
Pi/32 = Sum_{n >= 1} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9)).
Applying Euler's series transformation to this alternating sum gives
Pi/32 = Sum_{n >= 1} 2^(n-3)*n*(n+1)/((2*n+3)*binomial(2*n+2, n+1)). (End)
EXAMPLE
0.0981747704246810387019576057274844651311615437304720569054670185096...
MATHEMATICA
Join[{0}, RealDigits[Pi/32, 10, 105] // First]
PROG
(PARI) Pi/32 \\ Charles R Greathouse IV, Sep 28 2022
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved