|
%I #28 Sep 08 2022 08:45:51
%S 0,9,153,7641,192789,32757741,525987081,190358321841,23076404893161,
%T 577743530648769,578407918658769,556370890030917009,
%U 160916328686946575601,220439117509451225357769
%N a(n) = numerator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)), where Zeta is the Hurwitz Zeta function.
%C All numbers in this sequence are divisible by 9.
%H G. C. Greubel, <a href="/A173982/b173982.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+1)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.
%F A173982(n)/A173984(n) = sum_{i=0..n} 1/(1/3+i)^2 = 9*sum_{i=0..n} 1/(1+3i)^2 = psi'(1/3) - psi'(n+1/3). - _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 A173982 := proc(n) add( 1/(1/3+i)^2,i=0..n-1) ; numer(%) ; end proc: seq(A173982(n),n=0..20) ; # _R. J. Mathar_, Apr 22 2010
%t Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]], {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(sum(k=0,n-1, 9/(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 A173984.
%Y For A173982/9 see A173983.
%Y Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982-A173987.
%K frac,nonn
%O 0,2
%A _Artur Jasinski_, Mar 04 2010
%E Name simplified by _Peter Luschny_, Nov 14 2017
|
