Rationals r(n) = A121004(n)/A121005(n), n>=0.

 r(n):= rII(p=2,n) = sum(C(k)/((5^k)*F(2*2+1)^(2*k)),k=0..n), n>=0, with the Fibonacci number F(5)=5 and the Catalan

 numbers C(k):=A000108(k).

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

 [1, 126/125, 15752/15625, 393801/390625, 246125639/244140625, 30765704917/30517578125, 3845713114757/3814697265625, 480714139345054/476837158203125, 12017853483626636/11920928955078125, 7511158427266652362/7450580596923828125, 938894803408331562046/931322574615478515625, 117361850426041445314536/116415321826934814453125, 14670231303255180664525012/14551915228366851806640625, 73351156516275903322654776/72759576141834259033203125, 45844472822672439576659769888/45474735088646411895751953125, 5730559102834054947082473174969/5684341886080801486968994140625, 716319887854256868385309153942659/710542735760100185871124267578125, 89539985981782108548163644268761333/88817841970012523233890533447265625, 2238499649544552713704091106738138873/2220446049250313080847263336181640625, 1399062280965345446065056941711690248263/1387778780781445675529539585113525390625, 174882785120668180758132117713962593856959/173472347597680709441192448139190673828125, 21860348140083522594766514714245329125373279/21684043449710088680149056017398834228515625, 2732543517510440324345814339280666158968172603/2710505431213761085018632002174854278564453125, 68313587937761008108645358482016653987926699621/67762635780344027125465800054371356964111328125, 213479962305503150339516745256302043713560840462949/211758236813575084767080625169910490512847900390625, 26684995288187893792439593157037755464199967004270077/26469779601696885595885078146238811314105987548828125, 3335624411023486724054949144629719433025014242886831777/3308722450212110699485634768279851414263248443603515625, 416953051377935840506868643078714929128126849894404888129/413590306276513837435704346034981426782906055450439453125, 10423826284448396012671716076967873228203171300109712553297/10339757656912845935892608650874535669572651386260986328125, 6514891427780247507919822548104920767626982063570812562461993/6462348535570528709932880406796584793482907116413116455078125, 814361428472530938489977818513115095953372757950166556809841429/807793566946316088741610050849573099185363389551639556884765625]


 The numerators are A121004(n), n=0..30:

[1, 126, 15752, 393801, 246125639, 30765704917, 3845713114757, 480714139345054, 12017853483626636, 7511158427266652362, 938894803408331562046, 117361850426041445314536, 14670231303255180664525012, 73351156516275903322654776, 45844472822672439576659769888, 5730559102834054947082473174969, 716319887854256868385309153942659, 89539985981782108548163644268761333, 2238499649544552713704091106738138873, 1399062280965345446065056941711690248263, 174882785120668180758132117713962593856959, 21860348140083522594766514714245329125373279, 2732543517510440324345814339280666158968172603, 68313587937761008108645358482016653987926699621, 213479962305503150339516745256302043713560840462949, 26684995288187893792439593157037755464199967004270077, 3335624411023486724054949144629719433025014242886831777, 416953051377935840506868643078714929128126849894404888129, 10423826284448396012671716076967873228203171300109712553297, 6514891427780247507919822548104920767626982063570812562461993, 814361428472530938489977818513115095953372757950166556809841429]

 The denominators are A121005(n), n=0..30:

[1, 125, 15625, 390625, 244140625, 30517578125, 3814697265625, 476837158203125, 11920928955078125, 7450580596923828125, 931322574615478515625, 116415321826934814453125, 14551915228366851806640625, 72759576141834259033203125, 45474735088646411895751953125, 5684341886080801486968994140625, 710542735760100185871124267578125, 88817841970012523233890533447265625, 2220446049250313080847263336181640625, 1387778780781445675529539585113525390625, 173472347597680709441192448139190673828125, 21684043449710088680149056017398834228515625, 2710505431213761085018632002174854278564453125, 67762635780344027125465800054371356964111328125, 211758236813575084767080625169910490512847900390625, 26469779601696885595885078146238811314105987548828125, 3308722450212110699485634768279851414263248443603515625, 413590306276513837435704346034981426782906055450439453125, 10339757656912845935892608650874535669572651386260986328125, 6462348535570528709932880406796584793482907116413116455078125, 807793566946316088741610050849573099185363389551639556884765625]


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 For more details on this second p-family (here p=2) and the other three ones see the W. Lang link under A120996.

 This seond family has as limits of the series prime numbers in Q(sqrt(2)) (like the third family).

 The limits of this second p-family are (3-phi)*(1/phi)^(2*p)  showing up as (dimensionless) side lengths in the 

 golden triangle iteration. 

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 r(n) for n=10^k, k=0,1,2,3: (maple10, 15 digits):
 
 [1.00800000000000, 1.00813061875578, 1.00813061875578, 1.00813061875578]

 This should be compared with the limit of the series CsnII(2):=sum(C(k)/((5^k)*F(2*2+1)^(2*k),k=0..infinity) 

 with F(5) =5, which is 5*(18 - 11*phi) = 5*sqrt(5)/phi^5 = 1.0081306187560 (maple10, 15 digits). 

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