Rationals  r(n) = A120792(n)/A120793(n) = sum(C(k)/(-12)^k,k=0..n)

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

[1, 11/12, 67/72, 1603/1728, 9625/10368, 4277/4608, 230969/248832, 11086369/11943936, 199555357/214990848, 
2394661853/2579890176, 14367975317/15479341056, 344831378215/371504185344, 2068988321293/2229025112064, 
24827859669791/26748301344768, 49655719451017/53496602689536, 1588983021355339/1711891286065152, 
9533898130096349/10271347716390912, 343220332661861099/369768517790072832, 2059321996010969819/2218611106740436992, 
49423727903968731791/53246666561770487808, 296542367424359400781/319479999370622926848, 
3558508409090273953787/3833759992447475122176, 21351050454545455496207/23002559954684850733056, 
341616807272708229071887/368040959274957611728896, 2049700843636285205102081/2208245755649745670373376, 
24596410123635287407158265/26498949067796948044480512, 1328206146676307815905680329/1430943249661035194401947648, 
31876947520231370198348598895/34342637991864844665646743552, 191261685121388254158585562165/206055827951189067993880461312, 
2295140221456658924622749664559/2472669935414268815926565535744, 13770841328739953786173154368123/14836019612485612895559393214464]


The values of some partial sums r(n) of the convergent series sum(((-1)^k)*C(k)/12^k,k=0..infty) are (maple10 10 digits):

  [.9166666667, .9282032914, .9282032303, .9282032303]  for n=10^k with k=0..3.

  This should be compared with the limit 2*(2*sqrt(3)-3)) = 0.9282032302...

 The sum sum(C(k)/(-12)^k,k=0..infinity) is convergent due to Leibniz' criterion because {C(k)/12^k} is a monotonely decreasing 
 0-sequence. The latter fact follows from the convergence of sum({C(k)/12^k,k=0..infinity), which can be shown with 
 J. L. Raabe's criterion (cf. A120782(n)/A120783(n) and the W. Lang link there).
  

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