Rationals r(n) = A124397(n)/ A124398(n)
 
 r(n) = sum(((-1)^k)*binomial(2*k,k)/5^k),k=0..n)

 r(n) =  sum(((-1)^k)*((2*k-1)!!/(2*k)!!)*(4/5)^k,k=0..n)

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

  [1, 3/5, 21/25, 17/25, 99/125, 2223/3125, 12039/15625, 56763/78125, 

   59337/78125, 286961/390625, 7358781/9765625, 36088473/48828125, 

   183146521/244140625, 181066401/244140625, 36534213/48828125, 

   4535753121/6103515625, 22798981683/30517578125, 113528187171/152587890625, 

   113891192583/152587890625, 568042152363/762939453125, 

   14228623114839/19073486328125, 71035463999307/95367431640625, 

   355598139789279/476837158203125, 14210752102407/19073486328125, 

   1777633916948199/2384185791015625, 222077829012087123/298023223876953125, 

   1110885063593383719/1490116119384765625, 

   5552478378541270483/7450580596923828125, 

   5554008116661422571/7450580596923828125, 

   27764027130007204647/37252902984619140625, 

   694218942831744977599/931322574615478515625]


 Numerators are A124397(n); n=0..30:

  [1, 3, 21, 17, 99, 2223, 12039, 56763, 59337, 286961, 7358781, 36088473, 

  183146521, 181066401, 36534213, 4535753121, 22798981683, 113528187171, 

  113891192583, 568042152363, 14228623114839, 71035463999307, 355598139789279, 

  14210752102407, 1777633916948199, 222077829012087123, 1110885063593383719, 

  5552478378541270483, 5554008116661422571, 27764027130007204647, 

  694218942831744977599]

 
 Denominators are A124398(n); n=0..30: 

 [1, 5, 25, 25, 125, 3125, 15625, 78125, 78125, 390625, 9765625, 48828125, 

  244140625, 244140625, 48828125, 6103515625, 30517578125, 152587890625, 

  152587890625, 762939453125, 19073486328125, 95367431640625, 476837158203125,
 
  19073486328125, 2384185791015625, 298023223876953125, 1490116119384765625, 

  7450580596923828125, 7450580596923828125, 37252902984619140625, 

  931322574615478515625]


  Note that this is n o t A123748, the denominators of partial sums for 

  sqrt(5). Compare, e.g. n=3, 14, n=23,...
 
 
  The values for r(10^k), for k=0..3 are: (10 digits Maple 10)

  [.6000000000, .7535391744, .7453559925, .7453559925]

  This should be compared to 

  sqrt(5)/3 = (-1+2*phi)/3 = 0.7453559923 (10 digits Maple 10).

  phi:=(1+sqrt(5))/2 is the golden section number.


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