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a(n) = numerator of (Zeta(0,2,2/3) - Zeta(0,2,n+2/3)), where Zeta is the Hurwitz Zeta function.
4

%I #27 Sep 08 2022 08:45:51

%S 0,9,261,4401,546921,27234729,7956214281,8017899597,4266143013213,

%T 724241322449397,611292843754229277,9809672294036025657,

%U 9833902887524676921,3557459561652307818081,5990804897343048247416561

%N a(n) = numerator of (Zeta(0,2,2/3) - Zeta(0,2,n+2/3)), where Zeta is the Hurwitz Zeta function.

%C All numbers in this sequence are divisible by 9.

%H G. C. Greubel, <a href="/A173985/b173985.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = numerator of 2*(Pi^2)/3 - J - Zeta(2, (3*n+2)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.

%F a(n)/A173987(n) = sum_{i=0..n-1} 1/(i+2/3)^2 = psi'(2/3)-psi'(2/3+n). - _R. J. Mathar_, Apr 22 2010

%F a(n) = numerator of Sum_{k=0..(n-1)} 9/(3*k+1)^2. - _G. C. Greubel_, Aug 23 2018

%p A173985 := proc(n) add( 1/(2/3+i)^2,i=0..n-1) ; numer(%) ; end proc: seq(A173985(n),n=0..20) ; # _R. J. Mathar_, Apr 22 2010

%t Table[FunctionExpand[4*(Pi^2)/3 - Zeta[2, 1/3] - Zeta[2, (3*n + 2)/3]], {n, 0, 20}] // Numerator (* _Vaclav Kotesovec_, Nov 13 2017 *)

%t Numerator[Table[Sum[9/(3*k + 1)^2, {k, 0, n - 1}], {n, 0, 20}]] (* _G. C. Greubel_, Aug 23 2018 *)

%o (PARI) for(n=0,20, print1(numerator(9*sum(k=0,n-1, 1/(3*k+1)^2)), ", ")) \\ _G. C. Greubel_, Aug 23 2018

%o (Magma) [0] cat [Numerator((&+[9/(3*k+1)^2: k in [0..n-1]])): n in [1..20]]; // _G. C. Greubel_, Aug 23 2018

%Y For denominators see A173987.

%Y For A173985/9 see A173986.

%Y Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982, A173983, A173984.

%K frac,nonn

%O 0,2

%A _Artur Jasinski_, Mar 04 2010

%E Name simplified by _Peter Luschny_, Nov 14 2017