A173985 a(n) = numerator of (Zeta(0,2,2/3) - Zeta(0,2,n+2/3)), where Zeta is the Hurwitz Zeta function.

0, 9, 261, 4401, 546921, 27234729, 7956214281, 8017899597, 4266143013213, 724241322449397, 611292843754229277, 9809672294036025657, 9833902887524676921, 3557459561652307818081, 5990804897343048247416561

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+2)/3), where Zeta is the Hurwitz Zeta function and the constant J is A173973.

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

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

Maple

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

Mathematica

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 *)
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(9*sum(k=0, n-1, 1/(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 A173987.

For A173985/9 see A173986.

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

Sequence in context: A229259 A117796 A117051 * A280178 A189643 A003387

Adjacent sequences: A173982 A173983 A173984 * A173986 A173987 A173988

Keyword

frac,nonn

Author

Artur Jasinski, Mar 04 2010

Extensions

Name simplified by Peter Luschny, Nov 14 2017

Status

approved