OFFSET
1,2
COMMENTS
a((p-1)/2) is divisible by prime p > 3.
The limit of the rationals r(n) = Sum_{k=1..n} 1/(2*k-1)^2, for n -> infinity, is (Pi^2)/8 = (1 - 1/2^2)*Zeta(2), which is approximately 1.233700550.
r(n) = (Psi(1, 1/2) - Psi(1, n+1/2))/4 for n >= 1, where Psi(n,k) = Polygamma(n,k) is the n-th derivative of the digamma function. Psi(1, 1/2) = 3*Zeta(2) = Pi^2/2. - Jean-François Alcover, Dec 02 2013 [Corrected by Petros Hadjicostas, May 09 2020]
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..300
Wolfdieter Lang, Rationals and limit.
FORMULA
a(n) = numerator( Pi^2/2 - Zeta(2,(2n+1)/2) ) / 4 for n > 0. - Artur Jasinski, Mar 03 2010 [corrected by Bruno Berselli, Dec 02 2013]
EXAMPLE
MATHEMATICA
Numerator[Table[Sum[1/(2k-1)^2, {k, 1, n}], {n, 1, 25}]]
Table[(PolyGamma[1, 1/2] - PolyGamma[1, n+1/2])/4 // Numerator, {n, 1, 15}] (* Jean-François Alcover, Dec 02 2013 *)
Accumulate[1/(2*Range[20]-1)^2]//Numerator (* Harvey P. Dale, Jun 14 2020 *)
PROG
(PARI) for(n=1, 20, print1(numerator(sum(k=1, n, 1/(2*k-1)^2)), ", ")) \\ G. C. Greubel, Aug 23 2018
(Magma) [Numerator((&+[1/(2*k-1)^2: k in [1..n]])): n in [1..20]]; // G. C. Greubel, Aug 23 2018
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Alexander Adamchuk, Jul 01 2006
STATUS
approved