Rationals r(n):=A120782(n)/A120783(n) = sum(C(k)/12^k,k=0..n).

 For n=0..30:
  
 [1, 13/12, 79/72, 1901/1728, 11413/10368, 45659/41472, 273965/248832, 13150463/11943936, 236709049/214990848,
   2840511019/2579890176, 17043070313/15479341056, 409033716905/371504185344, 2454202353433/2229025112064,
   29450428426921/26748301344768, 58900856965277/53496602689536, 1884827423966069/1711891286065152, 
   11308964545760729/10271347716390912, 407122723668993709/369768517790072832, 2442736342053765479/2218611106740436992, 
   58625672209584915361/53246666561770487808, 351754033258056502201/319479999370622926848, 
   4221048399098716881997/3833759992447475122176, 25326290394596113065467/23002559954684850733056, 
   135073548771185622638299/122680319758319203909632, 2431323877881377038160141/2208245755649745670373376, 
   29175886534576659511988399/26498949067796948044480512, 1575497872867141909566507565/1430943249661035194401947648, 
   37811948948811423212983910561/34342637991864844665646743552, 
   226871693692868572246397432161/206055827951189067993880461312, 
   2722460324314422992237046267353/2472669935414268815926565535744, 
  16334761945886538191858933984887/14836019612485612895559393214464]


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


  [1.083333333, 1.101020402, 1.101020514, 1.101020514]  for r(10^k), k=0,...,3. 

 
  This series is convergent (due to the quotient criterion). The limit is 2*(3-sqrt(6))=  1.101020514..... 
  from the convergent Taylor expansion of sqrt(1+x) for the value x=-1/3  (the radius of convergence is R=1 
  due to the quotient criterion). 
  The Lagrange remainder sequence for all |x|<1 tends to zero because 0<=|R(n,x)| < (1/2)*(C(n)/4^n) |x|^{n+1}, and
  (C(k)/4^k)*|x|^(k+1) is a 0-sequence for |x|<1  because the power series having these coefficients has radius of 
  convergence R=1. 


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