More information on the rational numbers  A101631(n)/A101632(n), n>=1:

The series s:=lim(s(n),n->infty) with partial sums 

s(n):=120*sum(1/((5*k-1)*(5*k)*(5*k+1)),k=1..n) =  24*sum(1/((5*k-1)*k*(5*k+1)),k=1..n)

converges to s= -24*(gamma + Psi(1/5) + 5/2 + Pi*cot(Pi/5)/2) = 1.1954056019...   
   
The rational numbers s(n) are, for n=1..25, (in lowest terms):

[1, 37/33, 1069/924, 20575/17556, 1346153/1141140, 1214756107/1025884860, 20699705479/17440042620, 
850029466379/715041747420, 19572345658457/16445960190660, 137116980686111/115121721334620, 
411600123273343/345365164003860, 1482039573988769177/1242969225249892140, 456179332236626381/382452069307659120, 
32398234503565880731/27154096920843797520, 1199020509231104363863/1004701586071220508240, 
284223677442553468708211/238114275898879260452880, 12223619090331470408742593/10238913863651808199473840, 
14144677556820873512096032261/11846423340245142086791232880, 664877782166240562582242405957/556781896991521678079187945360, 
13431880974053796386411185540805/11246994319228737897199596496272, 
711951488296934882863843380896161/596090698919123108551578614302416, 
77608570609613264621432162705537101/64973886182184418832122068958963344, 
77613697563466510452634886770001629/64973886182184418832122068958963344, 
853800309474817775726754665141491535/714712748004028607153342758548596784, 
21346105605914997432503971609733215559/17867818700100715178833568963714919600]


The numerators give A101631 for n=1..25:

[1, 37, 1069, 20575, 1346153, 1214756107, 20699705479, 850029466379, 19572345658457, 137116980686111, 
411600123273343, 1482039573988769177, 456179332236626381, 32398234503565880731, 1199020509231104363863, 
284223677442553468708211, 12223619090331470408742593, 14144677556820873512096032261, 664877782166240562582242405957, 
13431880974053796386411185540805, 711951488296934882863843380896161, 77608570609613264621432162705537101, 
77613697563466510452634886770001629, 853800309474817775726754665141491535, 21346105605914997432503971609733215559]


The denominators give A101632 for n=1..25:

[1, 33, 924, 17556, 1141140, 1025884860, 17440042620, 715041747420, 16445960190660, 115121721334620, 
345365164003860, 1242969225249892140, 382452069307659120, 27154096920843797520, 1004701586071220508240, 
238114275898879260452880, 10238913863651808199473840, 11846423340245142086791232880, 556781896991521678079187945360, 
11246994319228737897199596496272, 596090698919123108551578614302416, 64973886182184418832122068958963344, 
64973886182184418832122068958963344, 714712748004028607153342758548596784, 17867818700100715178833568963714919600]


The values of the rationals s(n) are (10 digits maple9.5), for n=1..25:

[1., 1.121212121, 1.156926407, 1.171964001, 1.179656309, 1.184105697, 1.186906817, 1.188782990, 1.190100513, 1.191060897, 1.191782398, 1.192338108, 1.192775171, 1.193125096, 1.193409591, 1.193644003, 1.193839430, 1.194004059, 1.194144037, 1.194264049, 1.194367719, 1.194457884, 1.194536792, 1.194606241, 1.194667685]

The values of s(n) for  n=10^k, for k=0..4, (10 digits maple9.5) are:

[1., 1.191060897, 1.195358079, 1.19540510, 1.19540560]

to be compared with  1.1954056019...

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The limit s = -24*(gamma + Psi(1/5) + 5/2 + Pi*cot(Pi/5)/2) is also given by the series with partial sums 

S(n):= 24*sum(Zeta(2*k+1)/5^(2*k),k=1..n) with Euler's (or Riemann's)Zeta function, 
and S(n) has for n=10^k, with k=0..4, the values (10 digits maple9.5):

[1.153974627, 1.195405601, 1.195405602, 1.195405602, 1.195405602]

For this see the Abramowitz-Stegun ref. given in A101628, p.259, eq. 6.3.15 with z=1/5, together with p.258.   

lim(S(n),n->infty) can be rewritten, using the definition of Zeta(2*k+1)=sum(1/n^(2*k+1),n=1..infty) and 
interchanging limits, to yields the series lim(s(n),n->infty) given above.


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