A223709 Decimal expansion of (Pi-1)*(2*Pi-1)/12.

9, 4, 2, 8, 6, 9, 2, 3, 6, 7, 8, 4, 1, 1, 1, 4, 6, 0, 1, 9, 0, 0, 8, 7, 6, 5, 4, 1, 5, 9, 4, 8, 2, 8, 0, 1, 5, 0, 2, 9, 9, 0, 8, 8, 4, 6, 9, 6, 3, 5, 5, 3, 1, 5, 8, 2, 5, 1, 5, 5, 4, 1, 4, 6, 2, 6, 3, 8, 6, 7, 0, 2, 1, 6, 4, 9, 8, 1, 9, 5, 7, 5, 0, 9, 9, 5, 3

Offset: 0

Author

Bruno Berselli, Mar 26 2013

Keywords

• nonn
• cons

Comments

Let p = sum(sin(k)/k, k>=1) = (Pi-1)/2 (A096444) and q = sum(sin(k/2)/k, k>=1) = (2*Pi-1)/4, then A223709 = (2/3)*p*q.

This is the case h=1 in sum(sin(k/h)/k^3, k>=1) = (h*Pi-1)*(2h*Pi-1)/(12*h^3) = ((h*Pi-1)/(2h))*((2h*Pi-1)/(4h))*(2/(3h)), where (j*Pi-1)/(2j) = sum(sin(k/j)/k, k>=1) and 1/j is real but not an integer multiple of 2Pi.

Examples

			0.9428692367841114601900876541594828015029908846963553...
		

References

• Tom M. Apostol, Calculus, Vol. 1, John Wiley & Sons, 1967 (2nd ed.). This constant is the case s=1, t=3 in sum(sin(n*s)/n^t, n>=1), see p. 409.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for transcendental numbers

Crossrefs

Cf. A096418, A096444, A217909, A223710.

Programs

Mathematica

RealDigits[(Pi - 1) (2 Pi - 1)/12, 10, 90][[1]]

Formula

Equals sum(sin(k)/k^3, k>=1).