This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A173986 #39 Sep 08 2022 08:45:51
%S A173986 0,1,29,489,60769,3026081,884023809,890877733,474015890357,
%T A173986 80471258049933,67921427083803253,1089963588226225073,
%U A173986 1092655876391630769,395273284628034202009,665644988593672027490729
%N A173986 a(n) = numerator((Psi(1, 2/3) - Psi(1, n+2/3))/9), where Psi(1, z) is the Trigamma function.
%C A173986 a(n+1)/A173987(n+1) gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(3*k+2)^2. The limit n -> infinity is given in A294967 as the Hurwitz Zeta function or the Trigamma function (1/9)*Zeta(2, 2/3) = (1/9)*Psi(1, 2/3) = 0.3404306010 ... - _Wolfdieter Lang_, Nov 12 2017
%H A173986 G. C. Greubel, <a href="/A173986/b173986.txt">Table of n, a(n) for n = 0..300</a>
%F A173986 a(n) = numerator(r(n)) with r(n) = (1/9)*(4*(Pi^2)/3 - Zeta(2, 1/3) - Zeta(2, (3*n+2)/3)) = (1/9)*(Zeta(2, 2/3) - Zeta(2, (3*n+2)/3)) with the Hurwitz Zeta function Zeta(2, q). This becomes the formula given in the name. - _Wolfdieter Lang_, Nov 13 2017
%F A173986 a(n) = numerator of (1/9)*(2(Pi^2)/3 - J - Zeta(2, (3n+2)/3)) where J is the constant A173973 [which becomes the preceding formula].
%F A173986 a(n) = numerator of Sum_{k=0..(n-2)} 2/(3*k+2)^2. - _G. C. Greubel_, Aug 23 2018
%e A173986 The rationals a(n)/A173987(n) begin 0/1, 1/4, 29/100, 489/1600, 60769/193600, 3026081/9486400, 884023809/2741569600, 890877733/2741569600, ... - _Wolfdieter Lang_, Nov 12 2017
%p A173986 r := n -> (Psi(1, 2/3) - Psi(1, n+2/3))/9:
%p A173986 seq(numer(simplify(r(n))), n=0..14); # _Peter Luschny_, Nov 13 2017
%t A173986 Table[Numerator[FunctionExpand[(4*Pi^2/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3])/9]], {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 14 2017 *)
%t A173986 Numerator[Table[Sum[2/(3*k + 2)^2, {k, 0, n - 2}], {n, 1, 20}]] (* _G. C. Greubel_, Aug 23 2018 *)
%o A173986 (PARI) for(n=1,20, print1(numerator(sum(k=0,n-2, 2/(3*k+2)^2)), ", ")) \\ _G. C. Greubel_, Aug 23 2018
%o A173986 (Magma) [0] cat [Numerator((&+[2/(3*k+2)^2: k in [0..n-2]])): n in [2..20]]; // _G. C. Greubel_, Aug 23 2018
%Y A173986 For denominators see A173987.
%Y A173986 For 9*A173986 see A173985.
%Y A173986 Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984, A173985, A294967.
%K A173986 frac,nonn,easy
%O A173986 0,3
%A A173986 _Artur Jasinski_, Mar 04 2010
%E A173986 Name simplified by _Peter Luschny_, Nov 13 2017