OFFSET
1,3
COMMENTS
The denominators are given in A173954.
a(n+2)/A173954(n+2) = (Zeta(2, 3/4) - Zeta(2, n + 7/4))/16 gives, for n >= 0, the partial sum Sum_{k=0..n} 1/(4*n + 3). In the limit n -> infinity the series value is Zeta(2,3/4)/16, with the Hurwitz Zeta function, and it is given in A247037. - Wolfdieter Lang, Nov 15 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..250
FORMULA
a(n) = numerator of r(n) with r(n) = (Pi^2 - 8*Catalan - Zeta(2, n - 1/4))/16, with the Hurwitz Zeta function Z(2, z), and the Catalan constant is given in A006752. With Zeta(2, 3/4) = Pi^2 - 8*Catalan this is the formula given in the name.
Numerator of Sum_{k=0..n-2} 1/(4*k + 3)^2, n >= 2, with a(1) = 0. - G. C. Greubel, Aug 23 2018
MAPLE
r := n -> (Zeta(0, 2, 3/4) - Zeta(0, 2, n-1/4))/16:
seq(numer(simplify(r(n))), n=1..15); # Peter Luschny, Nov 14 2017
MATHEMATICA
Table[Numerator[FunctionExpand[(Pi^2 - 8*Catalan - Zeta[2, (4*n - 1)/4])/16]], {n, 1, 20}] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[8*n*Sum[(4*k - 1 + 2*n) / ((4*k - 1)^2 * (4*k - 1 + 4*n)^2), {k, 1, Infinity}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 14 2017 *)
Numerator[Table[Sum[1/(4*k + 3)^2, {k, 0, n-2}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 15 2017 *)
PROG
(PARI) for(n=1, 20, print1(numerator(sum(k=0, n-2, 1/(4*k+3)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018
(Magma) [0] cat [Numerator((&+[1/(4*k+3)^2: k in [0..n-2]])): n in [2..20]]; // G. C. Greubel, Aug 23 2018
CROSSREFS
KEYWORD
frac,nonn,easy
AUTHOR
Artur Jasinski, Mar 03 2010
EXTENSIONS
Numbers changed according to the old (or new) Mathematica program, and edited by Wolfdieter Lang, Nov 14 2017
Name simplified by Peter Luschny, Nov 14 2017
STATUS
approved