More information on the rational numbers  A101028(n)/A101029(n), n>=1:

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

s(n):=3*sum(1/(k*(2*k-1)*(2*k+1)),k=1..n) converges to s=3*(2*ln(2)-1)=1.1588830833...

(ln(2) = (s+3)/6 = .6931471805...)
   
The rational numbers s(n) are, for n=1..25, (in lowest terms):

[1, 11/10, 79/70, 479/420, 5297/4620, 69071/60060, 69203/60060, 471181/408408, 8960447/7759752, 44831407/38798760, 

1031626241/892371480, 5160071143/4461857400, 15484789693/13385572200, 64166447971/55454513400, 1989542332021/1719089915400, 

3979714828967/3438179830800, 27861681000449/24067258815600, 1030996803010973/890488576177200, 

1031094241305773/890488576177200, 42278288849598913/36510031623265200, 

1818093633186492859/1569931359800403600, 1818204269645957299/1569931359800403600, 

85460151199040573933/73786773910618969200, 598249092881851881881/516507417374332784400, 598273895158796627753/516507417374332784400]


The numerators give  A101028 for n=1..25:

[1, 11, 79, 479, 5297, 69071, 69203, 471181, 8960447, 44831407, 1031626241, 5160071143, 15484789693, 64166447971, 
1989542332021, 3979714828967, 27861681000449, 1030996803010973, 1031094241305773, 42278288849598913, 1818093633186492859, 
1818204269645957299, 85460151199040573933, 598249092881851881881, 598273895158796627753]

The denominators give A101029 for n=1..25:

[1, 10, 70, 420, 4620, 60060, 60060, 408408, 7759752, 38798760, 892371480, 4461857400, 13385572200, 55454513400, 
1719089915400, 3438179830800, 24067258815600, 890488576177200, 890488576177200, 36510031623265200, 1569931359800403600, 
1569931359800403600, 73786773910618969200, 516507417374332784400, 516507417374332784400]


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

[1., 1.100000000, 1.128571429, 1.140476190, 1.146536797, 1.150033300, 1.152231102, 1.153701690, 1.154733682, 1.155485562, 
1.156050215, 1.156484997, 1.156826878, 1.157100550, 1.157323020, 1.157506304, 1.157659093, 1.157787793, 1.157897214, 1.157991022, 
1.158072053, 1.158142525, 1.158204197, 1.158258474, 1.158306493]

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

[1., 1.155485562, 1.158845956, 1.158882704, 1.158883084]

#####################################################################################################################################

The limit s= 3*(2*ln(2)-1) is also given by the series with partial sums S(n):= 3*sum(Zeta(2*k+1)/2^(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):

[.9015426772, 1.158882129, 1.158883083, 1.158883083, 1.158883083]

For this see the Abramowitz-Stegun ref. given in A101028, p. 259, eq. 6.3.15 with z=1/2 together with p.258, eqs. 6.3.5 and 6.3.3.   

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.

#####################################################################################################################################