A128500 Partial sums of a series for Pi/(3*sqrt(3)).

 r(n):=sum(((-1)^k)*S(k,1)/(k+1)),k=0..n),
 with Chebyshev's S-Polynomials S(k,1)=[1,1,0,-1,-1,0], a 
 periodic sequence with period 6. See A010892.

 r(n) for n=0..35:

 [1, 1/2, 1/2, 3/4, 11/20, 11/20, 97/140, 159/280, 159/280, 187/280, 
  1777/3080, 1777/3080, 26181/40040, 23321/40040, 23321/40040, 51647/80080, 
  797919/1361360, 797919/1361360, 16521821/25865840, 15228529/25865840, 
  15228529/25865840, 16404249/25865840, 351431887/594914320, 
  351431887/594914320, 1876142299/2974571600, 1761735699/2974571600, 
  1761735699/2974571600, 1867970399/2974571600, 51196569971/86262576400, 
  51196569971/86262576400, 1673356245501/2674139868400, 
  3179578749227/5348279736800, 3179578749227/5348279736800, 
  3336881094427/5348279736800, 3184073101947/5348279736800, 
  3184073101947/5348279736800] 


 Numerators are this sequence (A128500), n=0..35:

 [1, 1, 1, 3, 11, 11, 97, 159, 159, 187, 1777, 1777, 26181, 23321, 23321, 
  51647, 797919, 797919, 16521821, 15228529, 15228529, 16404249, 351431887, 
  351431887, 1876142299, 1761735699, 1761735699, 1867970399, 51196569971, 
  51196569971, 1673356245501, 3179578749227, 3179578749227, 3336881094427, 
  3184073101947, 3184073101947]

 The entries between two repeated ones are a(3*k) = [1,3,97,187,26181,51647,16521821, 16404249,...]
 

 Denominators are in A128501, n=0..35:

 [1, 2, 2, 4, 20, 20, 140, 280, 280, 280, 3080, 3080, 40040, 40040, 40040, 
  80080, 1361360, 1361360, 25865840, 25865840, 25865840, 25865840, 594914320, 
  594914320, 2974571600, 2974571600, 2974571600, 2974571600, 86262576400, 
  86262576400, 2674139868400, 5348279736800, 5348279736800, 5348279736800, 
  5348279736800, 5348279736800]


 The entries between two repeated ones are A128501(3*k) = [1,4,140,280,40040,80080, 25865840,...]

 
 The values r(n) for n=10^k, k=0..5 are (maple10, 10 digits)

 [.5000000000, .5769480519, .6013320236, .6042671203, .6045664614, .6045964548]

  
 This shouls be compared with the approximate value for Pi/(3*sqrt(3)):

 0.6045997883.

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