A373506 Decimal expansion of 4*Pi/3^(3/2) - Pi^2/9.
1, 3, 2, 1, 7, 7, 6, 4, 4, 1, 0, 8, 0, 1, 3, 9, 5, 0, 9, 8, 1, 0, 4, 9, 4, 2, 3, 2, 4, 2, 5, 5, 2, 4, 1, 8, 3, 5, 6, 6, 1, 2, 1, 7, 2, 9, 9, 8, 5, 7, 8, 8, 4, 7, 5, 6, 0, 2, 8, 0, 7, 7, 6, 0, 9, 3, 7, 4, 9, 2, 5, 9, 4, 5, 6, 6, 3, 3, 7, 9, 2, 9, 0, 2, 3, 0, 8
Offset: 1
Examples
1.321776441080139509810494232425524183566...
Links
- Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27.
Crossrefs
Programs
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Maple
4*Pi/3^(3/2)-Pi^2/9 ; evalf(%) ;
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Mathematica
RealDigits[4*Pi/3^(3/2) - Pi^2/9, 10, 120][[1]] (* Amiram Eldar, Jun 10 2024 *)
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PARI
4*Pi/3^(3/2) - Pi^2/9 \\ Amiram Eldar, Jun 10 2024
Formula
Equals Sum_{n>=0} 1/((n+1)*binomial(2n,n)).
The alternating case is Sum_{n>=0} (-1)^n/((n+1)*binomial(2*n,n)) = 8*log(phi)/sqrt(5)-4*log^2(phi) = 0.79537... where phi is the golden ratio.