Rationals  r(n) = A120789(n)/A120781(n) = sum(C(k)/(-8)^k,k=0..n)
 
 Note: the denominators coincide with the numbers listed in A120781(n) but may differ for higher n values.
 C.f. the denominators A120787 and A120796 which are different. 

r(n), n=0..30:

[1, 7/8, 29/32, 459/512, 1843/2048, 14723/16384, 58925/65536, 1885171/2097152, 7541399/8388608, 60328761/67108864, 
241319243/268435456, 3861078495/4294967296, 15444365983/17179869184, 123554742139/137438953472, 
494219302861/549755813888, 31630025688259/35184372088832, 126520120431871/140737488355328, 
1012160898632573/1125899906842624, 4048643713939967/4503599627370496, 64778298539407877/72057594037927936, 
259113195798661613/288230376151711744, 2072905560272726149/2305843009213693952, 8291622252526225051/9223372036854775808,
265331911909309394807/295147905179352825856, 1061327647959713616059/1180591620717411303424, 
8490621182462222328109/9444732965739290427392, 33962484732144808446455/37778931862957161709568, 
543399755696933547414279/604462909807314587353088, 2173599022820702683625911/2417851639229258349412352, 
17388792182440341191925867/19342813113834066795298816, 69555168729999801424084237/77371252455336267181195264]


 The values of some partial sums r(n) of the convergent series sum(((-1)^k)*C(k)/8^k,k=0..infty) are (maple10 10 digits):

  [.8750000000, .8989842348, .8989794856, .8989794856] for n=10^k with k=0..3.

  This should be compared with the limit  2*(sqrt(6)-2)) = 0.898979485....

 The sum sum(C(k)/(-8)^k,k=0..infinity) is convergent due to Leibniz' criterion because {C(k)/8^k} is a monotonely 
 decreasing 0-sequence. The latter fact follows from the convergence of sum({C(k)/8^k,k=0..infinity), which can be 
 shown with J. L. Raabe's criterion (cf. A120780(n)/ A120781(n) and the W. Lang link there).
  

 ################################################################################################################################