A173982 a(n) = numerator of (Zeta(0,2,1/3) - Zeta(0,2,n+1/3)), where Zeta is the Hurwitz Zeta function.

0, 9, 153, 7641, 192789, 32757741, 525987081, 190358321841, 23076404893161, 577743530648769, 578407918658769, 556370890030917009, 160916328686946575601, 220439117509451225357769

Offset

0,2

Comments

All numbers in this sequence are divisible by 9.

Links

G. C. Greubel, Table of n, a(n) for n = 0..300

Formula

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.

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

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

Maple

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

Mathematica

Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]], {n, 0, 20}] // Numerator (* Vaclav Kotesovec, Nov 13 2017 *)
Numerator[Table[Sum[9/(3*k + 1)^2, {k, 0, n - 1}], {n, 0, 20}]] (* G. C. Greubel, Aug 23 2018 *)

Prog

(PARI) for(n=0, 20, print1(numerator(sum(k=0, n-1, 9/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018
(Magma) [0] cat [Numerator((&+[9/(3*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018

Crossrefs

For denominators see A173984.

For A173982/9 see A173983.

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982-A173987.

Sequence in context: A012148 A193540 A193543 * A185759 A215557 A365015

Adjacent sequences: A173979 A173980 A173981 * A173983 A173984 A173985

Keyword

frac,nonn

Author

Artur Jasinski, Mar 04 2010

Extensions

Name simplified by Peter Luschny, Nov 14 2017

Status

approved