A244978 Decimal expansion of Pi/32.

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

Author

Jean-François Alcover, Jul 09 2014

Keywords

• nonn
• cons
• easy

Examples

			0.0981747704246810387019576057274844651311615437304720569054670185096...
		

References

• George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), Chapter 13 A Master Formula, p. 250.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Eric Weisstein's MathWorld, Beta Function
• Eric W. Weisstein, Euler's Series Transformation.
• Index entries for transcendental numbers

Crossrefs

Cf. A019683, A244976, A244977, A019675, A000796, A003881, A019669, A019670, A019683.

Programs

Mathematica

Join[{0}, RealDigits[Pi/32, 10, 105] // First]

PARI

Pi/32 \\ Charles R Greathouse IV, Sep 28 2022

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)