Rationals r(n) = A120998(n)/A120999(n), n>=0.

   r(n):= rI(p=2,n) = sum(C(k)/L(2*2)^(2*k),k=0..n), n>=0, with the Lucas number L(4)=7 and the Catalan

   numbers C(k):=A000108(k).
  
   r(n), n=0..30:
  
  [1, 50/49, 2452/2401, 120153/117649, 841073/823543, 41212583/40353607, 14135916101/13841287201, 

  692659889378/678223072849, 33940334580952/33232930569601, 1663076394471510/1628413597910449, 

  81490743329120786/79792266297612001, 570435203303853900/558545864083284007, 

  27951324961888870816/27368747340080916343, 9587304461927883432788/9387480337647754305649, 

  469777918634466290881052/459986536544739960976801, 23019118013088848262866393/22539340290692258087863249, 

  1127936782641353564915810927/1104427674243920646305299201, 

  55268902349426324681004380213/54116956037952111668959660849, 

  386882316445984272767098895591/378818692265664781682717625943, 

  18957233505853229365588098350129/18562115921017574302453163671207, 

  6502331092507657672396724298214667/6366805760909027985741435139224001, 

  318614223532875225947439515078785703/311973482284542371301330321821976049, 

  15612096953110886071424536330343063087/15286700631942576193765185769276826401, 
  
  764992750702433417499802280529869704913/749048330965186233494494102694564493649, 
  
  37484644784419237457490311747253519688061/36703368217294125441230211032033660188801, 

  5354949254917033922498615963907534726287/5243338316756303634461458718861951455543, 

  1836747594436542635417025275620659255056689/1798465042647412146620280340569649349251249, 

  630004424891734123948039669537896057848860899/616873509628062366290756156815389726793178407, 

  30870216819694972073453943807356944512873005531/30226801971775055948247051683954096612865741943, 

  1512640624165053631599243246560490424308236792643/1481113296616977741464105532513750734030421355207, 

  74119390584087627948362919081464031336101674566979/72574551534231909331741171093173785967490646405143]


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 The numerators are A120998(n), n=0..30:

 [1, 50, 2452, 120153, 841073, 41212583, 14135916101, 692659889378, 33940334580952, 1663076394471510, 
 81490743329120786, 570435203303853900, 27951324961888870816, 9587304461927883432788, 469777918634466290881052, 
 23019118013088848262866393, 1127936782641353564915810927, 55268902349426324681004380213, 
 386882316445984272767098895591, 18957233505853229365588098350129, 6502331092507657672396724298214667, 
 318614223532875225947439515078785703, 15612096953110886071424536330343063087, 
 764992750702433417499802280529869704913, 37484644784419237457490311747253519688061, 
 5354949254917033922498615963907534726287, 1836747594436542635417025275620659255056689, 
 630004424891734123948039669537896057848860899, 30870216819694972073453943807356944512873005531, 
 1512640624165053631599243246560490424308236792643, 74119390584087627948362919081464031336101674566979]


 The denominators are A120999(n), n=0..30:

 [1, 49, 2401, 117649, 823543, 40353607, 13841287201, 678223072849, 33232930569601, 1628413597910449, 
 79792266297612001, 558545864083284007, 27368747340080916343, 9387480337647754305649, 459986536544739960976801, 
 22539340290692258087863249, 1104427674243920646305299201, 54116956037952111668959660849, 
 378818692265664781682717625943, 18562115921017574302453163671207, 6366805760909027985741435139224001, 
 311973482284542371301330321821976049, 15286700631942576193765185769276826401, 
 749048330965186233494494102694564493649, 36703368217294125441230211032033660188801, 
 5243338316756303634461458718861951455543, 1798465042647412146620280340569649349251249, 
 616873509628062366290756156815389726793178407, 30226801971775055948247051683954096612865741943, 
 1481113296616977741464105532513750734030421355207, 72574551534231909331741171093173785967490646405143]


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 For details on this p-family (here p=2) and the other three p-families see the W. Lang link under 

 A120996.

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 r(n) for n=10^k, k=0,1,2,3: (maple10, 15 digits):
 
 [1.02040816326531, 1.02128623625219, 1.02128623625221, 1.02128623625221]

 This should be compared with 

  the limit of the series CsnI(2):=sum(C(k)/L(2*2)^(2*k),k=0..infinity) with L(4) =7, is  

 7*(5 - 3*phi) = 7/phi^4 = 1.0212862362522 (maple10, 15 digits) 

 
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