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
KEYWORD
frac,nonn
AUTHOR
Artur Jasinski, Mar 04 2010
EXTENSIONS
Name simplified by Peter Luschny, Nov 14 2017
STATUS
approved