More information on the rational numbers  A101627(n)/A101628(n), n>=1:

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

s(n):=24*sum(1/((3*k-1)*(3*k)*(3*k+1)),k=1..n) converges to s= 12*(ln(3)-1) = 1.1833474640...

(ln(3) = s/12 +1 = 1.0986122886....)
   
The rational numbers s(n) are, for n=1..25, (in lowest terms):


[1, 39/35, 241/210, 34883/30030, 14039/12012, 1516871/1293292, 7601151/6466460, 
875425657/743642900, 7887002813/6692786100, 7095769757767/6016814703900, 
14199583385459/12033629407800, 75087685321529/63606326869800, 75113436870869/63606326869800, 
927229349730873529/784965679900201800, 927436191807263569/784965679900201800, 
305182576081725442901/258253708687166392200, 23479178371879154033/19865669899012799400, 
37713848011377144613/31905469837808435400, 37717984058802320713/31905469837808435400, 
135759786815564675620247/114827785946272559004600, 90513873992375059768273/76551857297515039336400, 
867274668206783459671292733/733443344767491591882048400, 
867328262897668917588127373/733443344767491591882048400, 
13486820603140621491169674571177/11404310567789726762173970571600, 
2697493899411257040501498771653/2280862113557945352434794114320]


The numerators give  A101627 for n=1..25:

[1, 39, 241, 34883, 14039, 1516871, 7601151, 875425657, 7887002813, 7095769757767, 
14199583385459, 75087685321529, 75113436870869, 927229349730873529, 927436191807263569, 
305182576081725442901, 23479178371879154033, 37713848011377144613, 37717984058802320713, 
135759786815564675620247, 90513873992375059768273, 867274668206783459671292733, 
867328262897668917588127373, 13486820603140621491169674571177, 2697493899411257040501498771653]


The denominators give A101628 for n=1..25:

[1, 35, 210, 30030, 12012, 1293292, 6466460, 743642900, 6692786100, 6016814703900, 
12033629407800, 63606326869800, 63606326869800, 784965679900201800, 784965679900201800, 
258253708687166392200, 19865669899012799400, 31905469837808435400, 31905469837808435400, 
114827785946272559004600, 76551857297515039336400, 733443344767491591882048400, 
733443344767491591882048400, 11404310567789726762173970571600, 2280862113557945352434794114320]


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

[1., 1.114285714, 1.147619048, 1.161605062, 1.168747919, 1.172875886, 1.175473288, 
1.177212419, 1.178433420, 1.179323298, 1.179991747, 1.180506547, 1.180911406, 1.181235528, 
1.181499033, 1.181716141, 1.181897137, 1.182049605, 1.182179239, 1.182290381, 1.182386387, 1.182469886, 1.182542959, 1.182607271, 1.182664170]

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

[1., 1.179323298, 1.183303462, 1.18334703, 1.18334747] 


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The limit s= 24*(ln(3)-1) is also given by the series with partial sums 

S(n):= 8*sum(Zeta(2*k+1)/3^(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):

[1.068495025, 1.183347464, 1.183347464, 1.183347464, 1.183347464]

For this see the Abramowitz-Stegun ref. given in A101627, p. 259, eq. 6.3.15 with z=1/3, 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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