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A244978 Decimal expansion of Pi/32.

%I A244978 #34 Feb 16 2025 08:33:23
%S A244978 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,
%T A244978 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,
%U A244978 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
%N A244978 Decimal expansion of Pi/32.
%D A244978 George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), Chapter 13 A Master Formula, p. 250.
%H A244978 Vincenzo Librandi, <a href="/A244978/b244978.txt">Table of n, a(n) for n = 0..10000</a>
%H A244978 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BetaFunction.html">Beta Function</a>
%H A244978 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/EulersSeriesTransformation.html">Euler's Series Transformation</a>.
%H A244978 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A244978 Equals Integral_{x = 0..1} x^2/(1 + x^2)^3 dx.
%F A244978 Also equals beta(3/2, 1/2)/16, where 'beta' is Euler's beta function.
%F A244978 From _Peter Bala_, Oct 27 2019: (Start)
%F A244978 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.
%F A244978 Equals Integral_{x >= 0} x^4/(1 + x^2)^4 dx. (End)
%F A244978 From _Amiram Eldar_, Jul 13 2020: (Start)
%F A244978 Equals Integral_{x=0..oo} dx/(x^2 + 4)^2.
%F A244978 Equals Sum_{k>=1} sin(k)^3*cos(k)^3/k. (End)
%F A244978 From  _Peter Bala_, Dec 08 2021: (Start)
%F A244978 Pi/32 = Sum_{n >= 1} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9)).
%F A244978 Applying Euler's series transformation to this alternating sum gives
%F A244978 Pi/32 = Sum_{n >= 1} 2^(n-3)*n*(n+1)/((2*n+3)*binomial(2*n+2, n+1)). (End)
%e A244978 0.0981747704246810387019576057274844651311615437304720569054670185096...
%t A244978 Join[{0}, RealDigits[Pi/32, 10, 105] // First]
%o A244978 (PARI) Pi/32 \\ _Charles R Greathouse IV_, Sep 28 2022
%Y A244978 Cf. A019683, A244976, A244977, A019675, A000796, A003881, A019669, A019670, A019683.
%K A244978 nonn,cons,easy
%O A244978 0,2
%A A244978 _Jean-François Alcover_, Jul 09 2014