A128506 Numerators of partial sums for a series for 3*sqrt(2)*(Pi^3)/2^7.

1, 28, 3473, 1187864, 32115203, 42776591068, 93938569006771, 93911487925744, 461478538827646397, 3165730339378740709148, 452199680641199918039, 5501473517781557885536888, 687727017229797976494536483

Offset: 0

Author

Wolfdieter Lang, Apr 04 2007

Keywords

• nonn
• frac
• easy

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.

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..385
• W. Lang, Rationals and limit.

Programs

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 *)

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

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.