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A173983 a(n) = numerator((Zeta(2, 1/3) - Zeta(2, n + 1/3))/9), where Zeta(n, z) is the Hurwitz Zeta function.

%I A173983 #31 Sep 08 2022 08:45:51
%S A173983 0,1,17,849,21421,3639749,58443009,21150924649,2564044988129,
%T A173983 64193725627641,64267546517641,61818987781213001,17879592076327397289,
%U A173983 24493235278827913928641,24506988360923903264741
%N A173983 a(n) = numerator((Zeta(2, 1/3) - Zeta(2, n + 1/3))/9), where Zeta(n, z) is the Hurwitz Zeta function.
%C A173983 From _Wolfdieter Lang_, Nov 12 2017: (Start)
%C A173983 a(n+1)/A173984(n+1) gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(1+3*k)^2.
%C A173983 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)
%H A173983 G. C. Greubel, <a href="/A173983/b173983.txt">Table of n, a(n) for n = 0..300</a>
%F A173983 a(n) = numerator of (1/9)(2(Pi^2)/3 + J - Zeta(2,(3n+1)/3)) where J is the constant A173973.
%F A173983 a(n) = numerator of Sum_{k=0..(n-1)} 1/(3*k+1)^2. - _G. C. Greubel_, Aug 23 2018
%e A173983 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
%p A173983 a := n -> numer((Zeta(0,2,1/3) - Zeta(0,2,n+1/3))/9):
%p A173983 seq(a(n), n=0..14); # _Peter Luschny_, Nov 12 2017
%t A173983 Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]]/9, {n, 0, 20}] // Numerator (* _Vaclav Kotesovec_, Nov 13 2017 *)
%t A173983 Numerator[Table[Sum[1/(3*k + 1)^2, {k, 0, n - 1}], {n, 0, 20}]] (* _G. C. Greubel_, Aug 23 2018 *)
%o A173983 (PARI) for(n=0,20, print1(numerator(sum(k=0,n-1, 1/(3*k+1)^2)), ", ")) \\ _G. C. Greubel_, Aug 23 2018
%o A173983 (Magma) [0] cat [Numerator((&+[1/(3*k+1)^2: k in [0..n-1]])): n in [1..20]]; // _G. C. Greubel_, Aug 23 2018
%Y A173983 For denominators see A173984.
%Y A173983 For 9*A173983 see A173982.
%Y A173983 Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982-A173987, A214550.
%K A173983 frac,nonn,easy
%O A173983 0,3
%A A173983 _Artur Jasinski_, Mar 04 2010
%E A173983 Name simplified by _Peter Luschny_, Nov 12 2017