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

Author

Artur Jasinski, Mar 04 2010

Keywords

• frac
• nonn

Comments

All numbers in this sequence are divisible by 9.

Links

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

Crossrefs

For denominators see A173984.

For A173982/9 see A173983.

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

Programs

Magma

[0] cat [Numerator((&+[9/(3*k+1)^2: k in [0..n-1]])): n in [1..20]]; // 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 *)

PARI

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

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

Extensions

Name simplified by Peter Luschny, Nov 14 2017