A173983 a(n) = numerator((Zeta(2, 1/3) - Zeta(2, n + 1/3))/9), where Zeta(n, z) is the Hurwitz Zeta function.

0, 1, 17, 849, 21421, 3639749, 58443009, 21150924649, 2564044988129, 64193725627641, 64267546517641, 61818987781213001, 17879592076327397289, 24493235278827913928641, 24506988360923903264741

Offset

0,3

Comments

From Wolfdieter Lang, Nov 12 2017: (Start)

a(n+1)/A173984(n+1) gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(1+3*k)^2.

The limit n -> infinity is given in A214550 as the Hurwitz Zeta function or the Polygamma function (1/9)*Zeta(2, 1/3) = (1/9)*Psi(1, 1/3) = 1.121733... (End)

Links

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

Formula

a(n) = numerator of (1/9)(2(Pi^2)/3 + J - Zeta(2,(3n+1)/3)) where J is the constant A173973.

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

Example

The rationals a(n)/A173984(n) begin 0/1, 1/1, 17/16, 849/784, 21421/19600, 3639749/3312400, 58443009/52998400, 21150924649/19132422400, ... - Wolfdieter Lang, Nov 12 2017

Maple

a := n -> numer((Zeta(0, 2, 1/3) - Zeta(0, 2, n+1/3))/9):
seq(a(n), n=0..14); # Peter Luschny, Nov 12 2017

Mathematica

Table[FunctionExpand[-Zeta[2, (3*n + 1)/3] + Zeta[2, 1/3]]/9, {n, 0, 20}] // Numerator (* Vaclav Kotesovec, Nov 13 2017 *)
Numerator[Table[Sum[1/(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, 1/(3*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018
(Magma) [0] cat [Numerator((&+[1/(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 9*A173983 see A173982.

Cf. A006752, A120268, A173945, A173947, A173948, A173949, A173953, A173955, A173973, A173982-A173987, A214550.

Sequence in context: A351769 A139091 A262634 * A231778 A355468 A281428

Adjacent sequences: A173980 A173981 A173982 * A173984 A173985 A173986

Keyword

frac,nonn,easy

Author

Artur Jasinski, Mar 04 2010

Extensions

Name simplified by Peter Luschny, Nov 12 2017

Status

approved