Rationals r(n) = A121012(n)/A121013(n), n>=0.

  r(n):= rIV(p=2,n) = sum(((-1)^k)*C(k)/L(2*2+1)^(2*k),k=0..n), n>=0, with the Lucas number L(5)=11  and the Catalan numbers C(k):=A000108(k).

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

  [1, 120/121, 14522/14641, 1757157/1771561, 212616011/214358881, 25726537289/25937424601, 282991910191/285311670611, 34242021133072/34522712143931, 4143284557101842/4177248169415651, 501337431409322440/505447028499293771, 667280121205808184436/672749994932560009201, 80740894665902790257970/81402749386839761113321, 9769648254574237621422382/9849732675807611094711841, 1182127438803482752191365322/1191817653772720942460132761, 143037420095221413015157878402/144209936106499234037676064081, 17307527831521790974834093591797/17449402268886407318558803753801, 2094210867614136707954925359965107/2111377674535255285545615254209921, 23036319543755503787504178947830287/23225154419887808141001767796309131, 2787394664794415958288005652730886427/2810243684806424785061213903353404851, 337274754440124330952848683980276597377/340039485861577398992406882305761986971, 40810245287255044045294690761614065020837/41144777789250865278081232758997200423491, 54318436477336463624287233403708296076467027/54763699237492901685126120802225273763666521, 6572530813757712098538755241848703916735073907/6626407607736641103900260617069258125403649041, 795276228464683163923189384263693173581884329097/801795320536133573571931534665380233173841533961, 96228423644226662834705915495906874004697907968061/97017233784872162402203715694511008214034825609281, 11643639260951426202999415775004731754563584917733929/11739085287969531650666649599035831993898213898723001, 1408880350575122570562929308775572542302212142398877561/1420429319844313329730664601483335671261683881745483121, 170474522419589831038114446361844277618567599696713268877/171871947701161912897410416779483616222663749691203457641, 1875219746615488141419258909980287053804243620640932480407/1890591424712781041871514584574319778449301246603238034051, 226901589340474065111730328107614733510313478006439901342759/228761562390246506066453264733492693192365450838991802120171,27455092310197361878519369701021382754747930839126045017209503/27680149049219827234040845032752615876276219551518008056540691]


 The numerators are A121012(n), n=0..30:

 [1, 120, 14522, 1757157, 212616011, 25726537289, 282991910191, 34242021133072, 4143284557101842, 501337431409322440, 667280121205808184436, 80740894665902790257970, 9769648254574237621422382, 1182127438803482752191365322, 143037420095221413015157878402, 17307527831521790974834093591797, 2094210867614136707954925359965107, 23036319543755503787504178947830287, 2787394664794415958288005652730886427, 337274754440124330952848683980276597377, 40810245287255044045294690761614065020837, 54318436477336463624287233403708296076467027, 6572530813757712098538755241848703916735073907, 795276228464683163923189384263693173581884329097, 96228423644226662834705915495906874004697907968061, 11643639260951426202999415775004731754563584917733929, 1408880350575122570562929308775572542302212142398877561, 170474522419589831038114446361844277618567599696713268877, 1875219746615488141419258909980287053804243620640932480407, 226901589340474065111730328107614733510313478006439901342759, 27455092310197361878519369701021382754747930839126045017209503]


  The denominators are A121013(n), n=0..30:


 [1, 121, 14641, 1771561, 214358881, 25937424601, 285311670611, 34522712143931, 4177248169415651, 505447028499293771, 672749994932560009201, 81402749386839761113321, 9849732675807611094711841, 1191817653772720942460132761, 144209936106499234037676064081, 17449402268886407318558803753801, 2111377674535255285545615254209921, 23225154419887808141001767796309131, 2810243684806424785061213903353404851, 340039485861577398992406882305761986971, 41144777789250865278081232758997200423491, 54763699237492901685126120802225273763666521, 6626407607736641103900260617069258125403649041, 801795320536133573571931534665380233173841533961, 97017233784872162402203715694511008214034825609281, 11739085287969531650666649599035831993898213898723001, 1420429319844313329730664601483335671261683881745483121, 171871947701161912897410416779483616222663749691203457641, 1890591424712781041871514584574319778449301246603238034051, 228761562390246506066453264733492693192365450838991802120171, 27680149049219827234040845032752615876276219551518008056540691]



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 For more details on this fourth p-family (here p=2 and normalized such that r(0)=1) and the other three ones see the 

 W. Lang link under A120996.

 This fourth family has as limits positive units in Q(sqrt(5)) (the other positive units are provided by the first family).

 The limits of this fourth p-family (unnormalized) are  (-F(2*p+2) + F(2*p+1))  = 1/phi^(2*p+1), p=1,2,3,... 

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


 r(n) for n=10^k, k=0,1,2,3: (maple10, 15 digits):
 
 [.991735537190083, .991869381244217, .991869381244217, .991869381244217]

 This should be compared with the value of the series CsnIV(2):=sum(((-1)^k)*C(k)/11^(2*k),k=0..infinity) which is 

 11*(-8 + 5*phi) = 11/phi^5 = 0.9918693836 (maple10, 15 digits).  


############################################## e.o.f. ########################################################################