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
KEYWORD
frac,nonn,easy
AUTHOR
Artur Jasinski, Mar 04 2010
EXTENSIONS
Name simplified by Peter Luschny, Nov 12 2017
STATUS
approved