More information on the rational numbers A101629(n)/A101630(n), n>=1: The series s:=lim(s(n),n->infty) with partial sums s(n):=60*sum(1/((4*k-1)*(4*k)*(4*k+1)),k=1..n) = 15*sum(1/((4*k-1)*k*(4*k+1)),k=1..n) converges to s= 15*(3*ln(2)-2) = 1.1916231251... (ln(2) = (s/15+2)/15 = .6931471805...) The rational numbers s(n) are, for n=1..25, (in lowest terms): [1, 47/42, 6931/6006, 238657/204204, 4563655/3879876, 526760263/446185740, 45934377581/38818159380, 2852342564497/2406725881560, 105651280880749/89048857617720, 4335127472172929/3651003162326520, 186521117762900387/156993135980040360, 61393482232562091673/51650741737433278440, 3255023127143379846869/2737489312083963757320, 3255958701070954680689/2737489312083963757320, 11720932854466449859557031/9852224034190185562594680, 23446376800523646141074957/19704448068380371125189360, 1571159220674321892105796919/1320198020581484865387687120, 8144418405772161604172168503777/6842586340673836057304382342960, 8145353824273962538698115787761/6842586340673836057304382342960, 1930638928195725617117489674542217/1621692962739699145581138615281520, 160256658712201022011890980286292091/134600515907395029083234505068366160, 14263897488363221641104404098967992699/11979445915758157588407870951084588240, 1097293895837418241032342347003417623/921495839673704429877528534698814480, 106443570368859001646123244031270317421/89385096448349329698120267865785004560, 2150268467022404377893747857029073484637/1805578948256656459902029410888857092112] The numerators give A101629 for n=1..25: [1, 47, 6931, 238657, 4563655, 526760263, 45934377581, 2852342564497, 105651280880749, 4335127472172929, 186521117762900387, 61393482232562091673, 3255023127143379846869, 3255958701070954680689, 11720932854466449859557031, 23446376800523646141074957, 1571159220674321892105796919, 8144418405772161604172168503777, 8145353824273962538698115787761, 1930638928195725617117489674542217, 160256658712201022011890980286292091, 14263897488363221641104404098967992699, 1097293895837418241032342347003417623, 106443570368859001646123244031270317421, 2150268467022404377893747857029073484637] The denominators give A101630 for n=1..25: [1, 42, 6006, 204204, 3879876, 446185740, 38818159380, 2406725881560, 89048857617720, 3651003162326520, 156993135980040360, 51650741737433278440, 2737489312083963757320, 2737489312083963757320, 9852224034190185562594680, 19704448068380371125189360, 1320198020581484865387687120, 6842586340673836057304382342960, 6842586340673836057304382342960, 1621692962739699145581138615281520, 134600515907395029083234505068366160, 11979445915758157588407870951084588240, 921495839673704429877528534698814480, 89385096448349329698120267865785004560, 1805578948256656459902029410888857092112] The values of the rationals s(n) are (10 digits maple9.5), for n=1..25: [1., 1.119047619, 1.154012654, 1.168718536, 1.176237333, 1.180585159, 1.183321886, 1.185154731, 1.186441732, 1.187379818, 1.188084540, 1.188627310, 1.189054187, 1.189395950, 1.189673805, 1.189902743, 1.190093604, 1.190254386, 1.190391092, 1.190508298, 1.190609543, 1.190697599, 1.190774661, 1.190842485, 1.190902491] The values of s(n) for n=10^k, for k=0..4, (10 digits maple9.5) are: [1., 1.187379818, 1.191576716, 1.19162273, 1.19162313] to be compared with 1.1916231251... ##################################################################################################################################### The limit s = 15*(3*ln(2)-2) is also given by the series with partial sums S(n):= 15*sum(Zeta(2*k+1)/4^(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): [1.126928347, 1.191623125, 1.191623126, 1.191623126, 1.191623126] For this see the Abramowitz-Stegun ref. given in A101627, p. 259, eq. 6.3.15 with z=1/4, together with p.258. lim(S(n),n->infty) can be rewritten, using the definition of Zeta(2*k+1)=sum(1/k^(2*k+1),k=1..infty) and interchanging limits, to yields the series lim(s(n),n->infty) given above. #####################################################################################################################################