Rationals r(n) = A121008(n)/A121009(n), n>=0.

   r(n):= rIII(p=2,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*2)^(2*k)),k=0..n), n>=0, with the Fibonacci number F(4)=3 

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

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

  [1, 44/45, 1982/2025, 17837/18225, 4013339/4100625, 60200071/61509375, 2709003239/2767921875, 121905145612/124556484375, 658287786362/672605015625, 740573759652388/756680642578125, 33325819184374256/34050628916015625, 1499661863296782734/1532278301220703125, 67484783848355431042/68952523554931640625, 607363054635198730798/620572711994384765625, 3036815273175993713422/3102863559971923828125, 136656687292919716888549/139628860198736572265625, 6149550928181387260770431/6283298708943145751953125, 830189375304487280195365199/848245325707324676513671875, 7471704377740385521764655307/7634207931365922088623046875, 1681133484991586742396929626529/1717696784557332469940185546875, 75651006824621403407862270801833/77296355305079961147308349609375, 3404295307107963153353800554998017/3478335988728598251628875732421875, 153193288819858341900921031073748341/156525119492786921323299407958984375, 459579866459575025702763091696535629/469575358478360763969898223876953125, 517027349767021903915608478301925265661/528272278288155859466135501861572265625, 23266230739515985676202381523046420687917/23772252522967013675976097583770751953125, 9422823449503974198861964516852167731678537/9627762271801640538770319521427154541015625, 424027055227678838948788403258278014374618161/433249302231073824244664378464221954345703125, 3816243497049109550539095629324554878961913521/3899243720079664418201979406177997589111328125, 858654786836049648871296516598023845524213890857/877329837017924494095445366390049457550048828125, 38639465407622234199208343246911076863576127180869/39479842665806602234295041487552225589752197265625] 


 The numerators are A121008(n), n=0..30:

 [1,44,1982,17837,4013339,60200071,2709003239,121905145612,658287786362,740573759652388,33325819184374256,1499661863296782734,67484783848355431042,607363054635198730798,3036815273175993713422,136656687292919716888549,6149550928181387260770431,830189375304487280195365199,7471704377740385521764655307,1681133484991586742396929626529,75651006824621403407862270801833,3404295307107963153353800554998017,153193288819858341900921031073748341,459579866459575025702763091696535629,517027349767021903915608478301925265661,23266230739515985676202381523046420687917,9422823449503974198861964516852167731678537,424027055227678838948788403258278014374618161,3816243497049109550539095629324554878961913521,858654786836049648871296516598023845524213890857,38639465407622234199208343246911076863576127180869]

  The denominators are A121009(n), n=0..30:

[1, 45, 2025, 18225, 4100625, 61509375, 2767921875, 124556484375, 672605015625, 756680642578125, 34050628916015625, 1532278301220703125, 68952523554931640625, 620572711994384765625, 3102863559971923828125, 139628860198736572265625, 6283298708943145751953125, 848245325707324676513671875, 7634207931365922088623046875, 1717696784557332469940185546875, 77296355305079961147308349609375, 3478335988728598251628875732421875, 156525119492786921323299407958984375, 469575358478360763969898223876953125, 528272278288155859466135501861572265625, 23772252522967013675976097583770751953125, 9627762271801640538770319521427154541015625, 433249302231073824244664378464221954345703125, 3899243720079664418201979406177997589111328125, 877329837017924494095445366390049457550048828125, 39479842665806602234295041487552225589752197265625]
 

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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 are (-4 + 3*phi)*(1/phi)^(2*(p-1)). Use (1/phi)^2 = 2 - phi.


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 r(n) for n=10^k, k=0,1,2,3: (maple10, 15 digits):
 
 [0.977777777777778, 0.978713763747827, 0.978713763747792, 0.978713763747792]

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

 3*(-11 + 7*phi) = 0.9787137637479 (maple10, 15 digits).  (maple10, 15 digits). 

 
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