Rationals r(n) = A121010(n)/A121011(n), n>=0.

   r(n):= rIII(p=3,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*3)^(2*k)),k=0..n), n>=0, with the Fibonacci number F(6)=8 

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

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

  [1, 319/320, 51041/51200, 6533247/6553600, 5226597607/5242880000, 1672511234219/1677721600000, 267601797475073/268435456000000, 342530300768093011/343597383680000000, 2192193924915795299/2199023255552000000, 17537551399326362389569/17592186044416000000000, 2806008223892217982335239/2814749767106560000000000, 1795845263291019508694523567/1801439850948198400000000000, 287335242126563121391123822723/288230376151711744000000000000, 18389455496100039769031924617127/18446744073709551616000000000000, 2942312879376006363045107938807181/2951479051793528258560000000000000, 7532320971202576289395476323344444391/7555786372591432341913600000000000000, 1205171355392412206303276211735114638327/1208925819614629174706176000000000000000, 385654833725571906017048387755236671300161/386856262276681335905976320000000000000000, 12340954679218300992545548408167573486381539/12379400392853802748991242240000000000000000, 39491054973498563176145754906136235156244198481/39614081257132168796771975168000000000000000000, 6318568795759770108183320784981797624999399962981/6338253001141147007483516026880000000000000000000, 2021942014643126434618662651194175239999806764840569/2028240960365167042394725128601600000000000000000000, 323510722342900229538986024191068038399969084661555131/324518553658426726783156020576256000000000000000000000, 82818744919782458761980422192913417830392085666496921263/83076749736557242056487941267521536000000000000000000000, 331274979679129835047921688771653671321568342666310161088831/332306998946228968225951765070086144000000000000000000000000, 106007993497321547215334940406929174822901869653218036061825557/106338239662793269832304564822427566080000000000000000000000000, 16961278959571447554453590465108667971664299144514888065811223139/17014118346046923173168730371588410572800000000000000000000000000, 10855218534125726434850297897669547501865151452489528344735795079959/10889035741470030830827987437816582766592000000000000000000000000000, 347366993092023245915209532725425520059684846479664907038139141352447/348449143727040986586495598010130648530944000000000000000000000000000,555787188947237193464335252360680832095495754367463851260897345886833779/557518629963265578538392956816209037649510400000000000000000000000000000, 88925950231557950954293640377708933135279320698794216201743813778549785409/89202980794122492566142873090593446023921664000000000000000000000000000000]


 The numerators are A121010(n), n=0..30:
 
 [1, 319, 51041, 6533247, 5226597607, 1672511234219, 267601797475073, 342530300768093011, 2192193924915795299, 17537551399326362389569, 2806008223892217982335239, 1795845263291019508694523567, 287335242126563121391123822723, 18389455496100039769031924617127, 2942312879376006363045107938807181, 7532320971202576289395476323344444391, 1205171355392412206303276211735114638327, 385654833725571906017048387755236671300161, 12340954679218300992545548408167573486381539, 39491054973498563176145754906136235156244198481, 6318568795759770108183320784981797624999399962981, 2021942014643126434618662651194175239999806764840569, 323510722342900229538986024191068038399969084661555131, 82818744919782458761980422192913417830392085666496921263, 331274979679129835047921688771653671321568342666310161088831, 106007993497321547215334940406929174822901869653218036061825557, 16961278959571447554453590465108667971664299144514888065811223139, 10855218534125726434850297897669547501865151452489528344735795079959, 347366993092023245915209532725425520059684846479664907038139141352447, 555787188947237193464335252360680832095495754367463851260897345886833779, 88925950231557950954293640377708933135279320698794216201743813778549785409]  


 The denominators are A121011(n), n=0..30:

[1, 320, 51200, 6553600, 5242880000, 1677721600000, 268435456000000, 343597383680000000, 2199023255552000000, 17592186044416000000000, 2814749767106560000000000, 1801439850948198400000000000, 288230376151711744000000000000, 18446744073709551616000000000000, 2951479051793528258560000000000000, 7555786372591432341913600000000000000, 1208925819614629174706176000000000000000, 386856262276681335905976320000000000000000, 12379400392853802748991242240000000000000000, 39614081257132168796771975168000000000000000000, 6338253001141147007483516026880000000000000000000, 2028240960365167042394725128601600000000000000000000, 324518553658426726783156020576256000000000000000000000, 83076749736557242056487941267521536000000000000000000000, 332306998946228968225951765070086144000000000000000000000000, 106338239662793269832304564822427566080000000000000000000000000, 17014118346046923173168730371588410572800000000000000000000000000, 10889035741470030830827987437816582766592000000000000000000000000000, 348449143727040986586495598010130648530944000000000000000000000000000, 557518629963265578538392956816209037649510400000000000000000000000000000, 89202980794122492566142873090593446023921664000000000000000000000000000000]



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 For more details on this third p-family (here p=1) and the other three ones see the W. Lang link under A120996.

 This third family has as limits of the series prime numbers in Q(sqrt(5)) (like the second family).

 The limits of this third p-family (unnormalized) are  (-4 + 3*phi)*(1/phi)^(2*(p-1)) = sqrt(5)/phi^(2*p). 

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


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 r(n) for n=10^k, k=0,1,2,3: (maple10, 15 digits):
 
 [.996875000000000, .996894379984858, .996894379984858, .996894379984858]

 This should be compared with the value of the series CsnIII(3):=sum(((-1)^k)*C(k)/((5^k)*8^(2*k)),k=0..infinity) which is 

 8*(-29 + 18*phi) = 8*sqrt(5)/phi^6 =  0.996894379985 (maple10, 15 digits).  


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