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A173983
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a(n) = numerator((Zeta(2, 1/3) - Zeta(2, n + 1/3))/9), where Zeta(n, z) is the Hurwitz Zeta function.
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10
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0, 1, 17, 849, 21421, 3639749, 58443009, 21150924649, 2564044988129, 64193725627641, 64267546517641, 61818987781213001, 17879592076327397289, 24493235278827913928641, 24506988360923903264741
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OFFSET
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0,3
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COMMENTS
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a(n+1)/A173984(n+1) gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(1+3*k)^2.
The limit n -> infinity is given in A214550 as the Hurwitz Zeta function or the Polygamma function (1/9)*Zeta(2, 1/3) = (1/9)*Psi(1, 1/3) = 1.121733... (End)
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LINKS
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FORMULA
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a(n) = numerator of (1/9)(2(Pi^2)/3 + J - Zeta(2,(3n+1)/3)) where J is the constant A173973.
a(n) = numerator of Sum_{k=0..(n-1)} 1/(3*k+1)^2. - G. C. Greubel, Aug 23 2018
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EXAMPLE
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The rationals a(n)/A173984(n) begin 0/1, 1/1, 17/16, 849/784, 21421/19600, 3639749/3312400, 58443009/52998400, 21150924649/19132422400, ... - Wolfdieter Lang, Nov 12 2017
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MAPLE
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a := n -> numer((Zeta(0, 2, 1/3) - Zeta(0, 2, n+1/3))/9):
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MATHEMATICA
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Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]]/9, {n, 0, 20}] // Numerator (* Vaclav Kotesovec, Nov 13 2017 *)
Numerator[Table[Sum[1/(3*k + 1)^2, {k, 0, n - 1}], {n, 0, 20}]] (* G. C. Greubel, Aug 23 2018 *)
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PROG
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(PARI) for(n=0, 20, print1(numerator(sum(k=0, n-1, 1/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018
(Magma) [0] cat [Numerator((&+[1/(3*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018
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CROSSREFS
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Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982-A173987, A214550.
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KEYWORD
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frac,nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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