Rationals r(n) = A121498(n)/A121499(n), n>=0.

   r(n):= rIV(p=3,n) = sum(((-1)^k)*C(k)/L(2*3+1)^(2*k),k=0..n), n>=0, with the Lucas number L(7)=29 

  and the Catalan numbers C(k):=A000108(k).

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

  [1, 840/841, 706442/707281, 594117717/594823321, 499653000011/500246412961, 420208173009209/420707233300201, 353395073500744901/353814783205469041, 297205256814126461312/297558232675799463481, 249949620980680353964822/250246473680347348787521, 210207631244752177684410440/210457284365172120330305161, 176784617876836581432589196836/176994576151109753197786640401, 148675863634419564984807514480290/148852438543083302439338564577241, 125036401316546854152223119678131902/125184900814733057351483732809459681, 105155613507215904342019643649308186682/105280501585190501232597819292755591721, 88435870959568575551638520309068187674002/88540901833145211536614766025207452637361, 2564640257827488690997517088962977442211753/2567686153161211134561828214731016126483469, 2156862456832917989128911871817864028901303503/2159424054808578564166497528588784562372597429, 1813921326196484028857414884198823648305991775513/1816075630094014572464024421543167816955354437789, 1525507835331243068269085917611210688225339099676733/1527319604909066255442244538517804134059453082180549, 1282952089513575420414301256711028188797510182767192343/1284475787728524720826927656893473276744000042113841709, 1078962707280916928568427356893974706778706063707435109443/1080244137479689290215446159447411025741704035417740877269, 907407636823251136926047407147832728400891799577952083377183/908485319620418693071190220095272672648773093786320077783229, 763129822568354206154805869411327324585150003445057705274782063/764036153800772120872870975100124317697618171874295185415695589, 641792180779985887376191736174926279976111152897293530124262073133/642554405346449353654084490059204551183696882546282250934599990349, 539747224035968131283377250123113001459909479586623858834548882958209/540388254896363906423085056139791027545489078221423373035998591883509, 453927415414249198409320267353538034227783872332350665279855442914529581/454466522367842045301814532213564254165756314784217056723274815774031069, 381752956363383575862238344844325486785566236631506909500358428124476380109/382206345311355160098826021591607537753401060733526544704274120065960129029, 321054236301605587300142448014077734386661205007097310889801438050286927019393/321435536406849689643112684158541939250610292076895824096294534975472468513389, 7830191769159858668663174164615341863956280128918096315291367272608711611027726237/7839491297426657080705875253942679356383134413463412253884527413516798034573044321, 6585191277863441140345729472441502507587231588420119001160039876263925462632101113949/6593012181135818604873641088565793338718216041722729705516887554767627147075930273961, 5538145864683153999030758486323303608880861765861320079975593535937961317888583538923413/5544723244335223446698732155483832197862019691088815682339702433559574430690857360401201] 


 The numerators are A121498(n), n=0..30:

 [1, 840, 706442, 594117717, 499653000011, 420208173009209, 353395073500744901, 297205256814126461312, 249949620980680353964822, 210207631244752177684410440, 176784617876836581432589196836, 148675863634419564984807514480290, 125036401316546854152223119678131902, 105155613507215904342019643649308186682, 88435870959568575551638520309068187674002, 2564640257827488690997517088962977442211753, 2156862456832917989128911871817864028901303503, 1813921326196484028857414884198823648305991775513, 1525507835331243068269085917611210688225339099676733, 1282952089513575420414301256711028188797510182767192343, 1078962707280916928568427356893974706778706063707435109443, 907407636823251136926047407147832728400891799577952083377183, 763129822568354206154805869411327324585150003445057705274782063, 641792180779985887376191736174926279976111152897293530124262073133, 539747224035968131283377250123113001459909479586623858834548882958209, 453927415414249198409320267353538034227783872332350665279855442914529581, 381752956363383575862238344844325486785566236631506909500358428124476380109, 321054236301605587300142448014077734386661205007097310889801438050286927019393, 7830191769159858668663174164615341863956280128918096315291367272608711611027726237, 6585191277863441140345729472441502507587231588420119001160039876263925462632101113949, 5538145864683153999030758486323303608880861765861320079975593535937961317888583538923413]

 The denominators are A121499(n), n=0..30: 

[1, 841, 707281, 594823321, 500246412961, 420707233300201, 353814783205469041, 297558232675799463481, 250246473680347348787521, 210457284365172120330305161, 176994576151109753197786640401, 148852438543083302439338564577241, 125184900814733057351483732809459681, 105280501585190501232597819292755591721, 88540901833145211536614766025207452637361, 2567686153161211134561828214731016126483469, 2159424054808578564166497528588784562372597429, 1816075630094014572464024421543167816955354437789, 1527319604909066255442244538517804134059453082180549, 1284475787728524720826927656893473276744000042113841709, 1080244137479689290215446159447411025741704035417740877269, 908485319620418693071190220095272672648773093786320077783229, 764036153800772120872870975100124317697618171874295185415695589, 642554405346449353654084490059204551183696882546282250934599990349, 540388254896363906423085056139791027545489078221423373035998591883509, 454466522367842045301814532213564254165756314784217056723274815774031069, 382206345311355160098826021591607537753401060733526544704274120065960129029, 321435536406849689643112684158541939250610292076895824096294534975472468513389, 7839491297426657080705875253942679356383134413463412253884527413516798034573044321, 6593012181135818604873641088565793338718216041722729705516887554767627147075930273961, 5544723244335223446698732155483832197862019691088815682339702433559574430690857360401201]


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

 For more details on this fourth p-family (here p=3 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)*phi =  1/phi^(2*p+1), p=1,2,3,... 

 Use (1/phi)^2 = 2 - phi.


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


 r(n) for n=10^k, k=0,1,2,3: (maple10, 15 digits):
 
 [0.998810939357907, 0.998813758710358, 0.998813758710358, 0.998813758710358]

 This should be compared with the value of the series CsnIV(3):=sum(((-1)^k)*C(k)/29^(2*k),k=0..infinity) which is 

 29*(-21 + 13*phi) = 29/phi^7 = 0.998813758709 (maple10, 15 digits).  


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