OFFSET
0,2
COMMENTS
The denominators are given in A128507.
The limit n -> infinity of the rationals r(n) defined below is 3*sqrt(2)*(Pi^3)/2^7 = 1.027756...
This series is obtained from the Fourier series for y(x)= x*(Pi-x) if 0<=x<=Pi and y(x)= (Pi-x)*(2*Pi-x) if Pi<=x<=2*Pi evaluated at x=Pi/4.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..385
W. Lang, Rationals and limit.
FORMULA
a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} S(2*k,sqrt(2))/(2*k+1)^3 with Chebyshev's S-Polynomials S(2*k,sqrt(2))=[1,1,-1,-1] periodic sequence with period 4. See A057077.
EXAMPLE
Rationals r(n): [1, 28/27, 3473/3375, 1187864/1157625, 32115203/31255875,...].
3*sqrt(2)*(Pi^3)/2^7 = 1/1^3 + 1/3^3 - 1/5^3 - 1/7^3 + 1/9^3 + 1/11^3 - 1/13^3 - 1/15^3 + ...
MATHEMATICA
r[n_]:= Sum[(1/2)*((1-I)*I^k+(1+I)*(-I)^k)/(2k+1)^3, {k, 0, n}]; Numerator[Table[r[n], {n, 0, 30}]] (* G. C. Greubel, Mar 28 2018 *)
PROG
(PARI) {r(n) = sum(k=0, n, ((1-I)*I^k + (1+I)*(-I)^k)/(2*(2*k+1)^3))};
for(n=0, 30, print1(numerator(r(n)), ", ")) \\ G. C. Greubel, Mar 28 2018
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Apr 04 2007
STATUS
approved