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A247669
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Decimal expansion of A_3 = Sum_{n >= 1} H(n)^2/((2n-1)*(2n)*(2n+1))^3, where H(n) is the n-th harmonic number.
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2
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0, 0, 4, 6, 4, 0, 4, 5, 0, 2, 0, 3, 5, 7, 8, 1, 2, 3, 9, 8, 9, 1, 8, 0, 5, 2, 9, 2, 3, 6, 2, 3, 4, 5, 4, 6, 7, 2, 5, 2, 6, 9, 3, 9, 0, 7, 8, 3, 2, 0, 2, 7, 3, 9, 2, 3, 0, 0, 4, 9, 2, 4, 9, 0, 7, 2, 3, 4, 4, 9, 7, 1, 8, 3, 2, 4, 1, 2, 5, 7, 3, 5, 3, 7, 4, 0, 5, 9, 7, 0, 3, 7, 2, 2, 8, 3, 4, 3, 8, 7, 8, 1, 0, 5
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OFFSET
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0,3
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LINKS
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FORMULA
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A_3 = 4*log(2)^3 + ((7*zeta(3))/8 - 35/4)*log(2)^2 - ((-(1/8))*9*zeta(2) + (7*zeta(3))/8 + (45*zeta(4))/32 - 12)*log(2) - zeta(2)/4 - (41*zeta(3))/8 - (3/32)*zeta(2)*zeta(3) + (45*zeta(4))/64 + (17*zeta(5))/32.
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EXAMPLE
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0.00464045020357812398918052923623454672526939078320273923...
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MATHEMATICA
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A3 = 4*Log[2]^3 + ((7*Zeta[3])/8 - 35/4)*Log[2]^2 - ((-(1/8))*9*Zeta[2] + (7*Zeta[3])/8 + (45*Zeta[4])/32 - 12)*Log[2] - Zeta[2]/4 - (41*Zeta[3])/8 - (3/32)*Zeta[2]*Zeta[3] + (45*Zeta[4])/64 + (17*Zeta[5])/32; Join[{0, 0}, RealDigits[A3, 10, 102] // First]
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PROG
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(PARI) 4*log(2)^3 + ((7*zeta(3))/8 - 35/4)*log(2)^2 - ((-(1/8))*9*zeta(2) + (7*zeta(3))/8 + (45*zeta(4))/32 - 12)*log(2) - zeta(2)/4 - (41*zeta(3))/8 - (3/32)*zeta(2)*zeta(3) + (45*zeta(4))/64 + (17*zeta(5))/32 \\ Michel Marcus, Sep 22 2014
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); 4*Log(2)^3 + ((7*Evaluate(L, 3))/8 - 35/4)*Log(2)^2 - ((-(1/8))*9*Evaluate(L, 2) + (7*Evaluate(L, 3))/8 + (45*Evaluate(L, 4))/32 - 12)*Log(2) - Evaluate(L, 2)/4 - (41*Evaluate(L, 3))/8 - (3/32)*Evaluate(L, 2)*Evaluate(L, 3) + (45*Evaluate(L, 4))/64 + (17*Evaluate(L, 5))/32; // G. C. Greubel, Aug 31 2018
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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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