Rationals r(n) = A121002(n)/A121003(n), n>=0.

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

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

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

 [1, 6/5, 32/25, 33/25, 839/625, 4237/3125, 21317/15625, 107014/78125, 4292/3125, 2687362/1953125, 13453606/9765625, 67326816/48828125, 336842092/244140625, 336990672/244140625, 1685488248/1220703125, 8429380209/6103515625, 42153972579/30517578125, 210795791853/152587890625, 210814897401/152587890625, 5270725887663/3814697265625, 26354942262399/19073486328125, 131779604565399/95367431640625, 658916319339723/476837158203125, 658930041724269/476837158203125, 82367545119680949/59604644775390625, 411842587544806197/298023223876953125, 2059231305077103137/1490116119384765625, 10296226058936431689/7450580596923828125, 10296278808526781761/7450580596923828125, 257407972455386195393/186264514923095703125, 1287043677263433069269/931322574615478515625] 

  The numerators are A121002(n), n=0..30:

 [1, 6, 32, 33, 839, 4237, 21317, 107014, 4292, 2687362, 13453606, 67326816, 336842092, 336990672, 1685488248, 8429380209, 42153972579, 210795791853, 210814897401, 5270725887663, 26354942262399, 131779604565399, 658916319339723, 658930041724269, 82367545119680949, 411842587544806197, 2059231305077103137, 10296226058936431689, 10296278808526781761, 257407972455386195393, 1287043677263433069269]  

 The denominators are A121003(n), n=0..30:

 [1, 5, 25, 25, 625, 3125, 15625, 78125, 3125, 1953125, 9765625, 48828125, 244140625, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 152587890625, 3814697265625, 19073486328125, 95367431640625, 476837158203125, 476837158203125, 59604644775390625, 298023223876953125, 1490116119384765625, 7450580596923828125, 7450580596923828125, 186264514923095703125, 931322574615478515625]


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 For more details on this second p-family (here p=0) 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 squares of (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):
 
 [1.20000000000000, 1.37764925440000, 1.38196601124968, 1.38196601125011]

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

 with F(1) =1, which is 3 - phi = 1.38196601125010 (maple10, 15 digits). 

 
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