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A223709
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Decimal expansion of (Pi-1)*(2*Pi-1)/12.
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3
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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
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OFFSET
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0,1
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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Equals sum(sin(k)/k^3, k>=1).
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EXAMPLE
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0.9428692367841114601900876541594828015029908846963553...
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MATHEMATICA
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RealDigits[(Pi - 1) (2 Pi - 1)/12, 10, 90][[1]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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