A128506/A128507 

Partial sums of a series for 3*sqrt(2)*(Pi^3)/2^7.
Obtained from a Fourier series involving S(2*n,sqrt(2)) Chebyshev polynomials. 

Rationals r(n):=sum(S(2*k,sqrt(2))/(2*k+1)^3,k=0..n) with Chebyshev's S-Polynomials 
S(2*k,sqrt(2))=[1,1,-1,-1] periodic sequence with period 4. See A057077. 

r(n)= A128506(n)/ A128507(n)


The numerators of r(n) give A128506:

 [1, 28, 3473, 1187864, 32115203, 42776591068, 93938569006771, 93911487925744, 461478538827646397, 3165730339378740709148, 452199680641199918039, 5501473517781557885536888, 687727017229797976494536483, 18569547407314954333168531916, 452874623862445241518124074258699, 13491147262581262356033315736366698784, 13491512557676907586845031656371495659, 94442731186979529052352876291288205988, 4783715769545108666471065797662134800832039, 4783637301099030591162216652553687040972664, 329697721098799192722736756349340677217975241169,...]


The denominators of r(n) give A128507:

 [1, 27, 3375, 1157625, 31255875, 41601569625, 91398648466125, 91398648466125, 449041559914072125, 3079976059450620705375, 439996579921517243625, 5353438387905100303185375, 669179798488137537898171875, 18067854559179713523250640625, 440656904843834033118559874203125, 13127609852202659680635017212385296875, 13127609852202659680635017212385296875, 91893268965418617764445120486697078125, 4654669752905349245622438688012667098265625, 4654669752905349245622438688012667098265625, 320804494039989575357544096816521029079565140625,...]



The rationals r(n), n=0..20, are:

 [1, 28/27, 3473/3375, 1187864/1157625, 32115203/31255875, 42776591068/41601569625, 
93938569006771/91398648466125, 93911487925744/91398648466125, 
461478538827646397/449041559914072125, 3165730339378740709148/3079976059450620705375, 
452199680641199918039/439996579921517243625, 5501473517781557885536888/5353438387905100303185375, 
687727017229797976494536483/669179798488137537898171875, 
18569547407314954333168531916/18067854559179713523250640625, 
452874623862445241518124074258699/440656904843834033118559874203125, 
13491147262581262356033315736366698784/13127609852202659680635017212385296875, 
13491512557676907586845031656371495659/13127609852202659680635017212385296875, 
94442731186979529052352876291288205988/91893268965418617764445120486697078125, 
4783715769545108666471065797662134800832039/4654669752905349245622438688012667098265625, 
4783637301099030591162216652553687040972664/4654669752905349245622438688012667098265625, 
329697721098799192722736756349340677217975241169/320804494039989575357544096816521029079565140625]


In the limit n->infinity r(n) becomes  3*sqrt(2)*(Pi^3)/2^7 = 1.027722586 (maple10, 10 digits).   
 
Values r(10^k), k=0..4 (maple10, 10 digits):

[1.037037037, 1.027734535, 1.027722584, 1.027722586] 


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