OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
Eric Weisstein's World of Mathematics, Polygamma Function
FORMULA
a(n) = numerator of expression (8*Catalan + Pi^2 - Zeta(2, (4*n + 1)/4))/16.
a(n) = numerator(r(n)) with r(n) = (Zeta(2,1/4) - Zeta(2, n + 1/4))/16, with the Hurwitz Zeta function Z(2, k). With Zeta(2,1/4) = 8 Catalan + Pi^2 this is the preceding formula, and Zeta(2, n + 1/4) = Psi(1, n + 1/4) with the polygamma (trigamma) function Psi(1, k). - Wolfdieter Lang, Nov 14 2017
EXAMPLE
The rationals r(n) begin: 0/1, 1/1, 26/25, 2131/2025, 362164/342225, 105007621/98903025, 5156362654/4846248225, 129102916279/121156205625, 108696708796264/101892368930625, 13163623138673569/12328976640605625, ... - Wolfdieter Lang, Nov 14 2017
MAPLE
r := n -> (Psi(1, 1/4) - Zeta(0, 2, n+1/4))/16:
seq(numer(simplify(r(n))), n=0..13); # Peter Luschny, Nov 14 2017
MATHEMATICA
Table[Numerator[FunctionExpand[(8*Catalan + Pi^2 - Zeta[2, (4*n + 1)/4])/16]], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[Sum[1/(4*k + 1)^2, {k, 0, n-1}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
PROG
(PARI) for(n=0, 20, print1(numerator(sum(k=0, n-1, 1/(4*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018
(Magma) [0] cat [Numerator((&+[1/(4*k+1)^2: k in [0..n-1]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Artur Jasinski, Mar 03 2010
EXTENSIONS
Edited by Wolfdieter Lang, Nov 14 2017
Name changed according to a formula of Lang by Peter Luschny, Nov 14 2017
STATUS
approved