A079269, A079278, A080581, A080582

[Pfoertner, Hugo]

Leroy,

your proposed sequences with the numerators and denominators
of a[k+1] = a[k] + 1/(m + 1 /a[k]) with m=1 were put into EIS
by Neil. (A079269, A079278) He also calculated the ratios of successive
terms
(A080581, A080582)

Here is the corresponding information for m=1..10

 M =            1
           k   n(k)/n(k-1) (A080581)
           |   |   d(k)/d(k-1) (A080582)
           2   3   2
           3   7   5
           4   41   31
           5   1481   1171
           6   2001161   1638151
           7   3741092827721   3146427633211
           8   13288522158502275517773641   11417451157987217266902031
  M =            2
           2   4   3
           3   14   11
           4   178   145
           5   29506   24721
           6   824811586   706521601
           7   652329111297234946   568754681712768961
           8  4.115638988851026107045833898795825E+0035
              3.640305507874634375094701230112909E+0035
  M =            3
           2   5   4
           3   23   19
           4   497   421
           5   235457   203461
           6   53392510817   46882572661
           7   2766001621753572646817   2460798973136286293701
           8  7.464467658097380439898886826586278E+0042
              6.713425293781487374296340087251891E+0042
  M =            4
           2   6   5
           3   34   29
           4   1106   961
           5   1181186   1041841
           6   1356366308546   1211190974401
           7   1797577807670478066872066   1621742753248404750649921
           8  3.169450041902175265872704801018189E+0048
              2.884290816626138999205533398150123E+0048
  M =            5
           2   7   6
           3   47   41
           4   2137   1891
           5   4445737   3980551
           6   19331781443977   17480084846491
           7   366860213219411004186536137   334492399585397030860014511
           8  1.324910653244991827929474701923162E+0053
              1.216642776727249188586551648077548E+0053
  M =            6
           2   8   7
           3   62   55
           4   3746   3361
           5   13736066   12442081
           6   185330714795906   169230848613121
           7   33829062464554283053173648386   31104468447883888447638326401
           8  1.129558642119363274318103466198450E+0057
              1.044811593494545756135351564480421E+0057
  M =            7
           2   9   8
           3   79   71
           4   6113   5545
           5   36723521   33573961
           6   1328777538250241   1223034333643081
           7   1743286495517187276136090026881
1613958925733185372749832966921
           8  3.005596564840124448760657787207332E+0060
              2.796867179243853172114507012401849E+0060
  M =            8
           2   10   9
           3   98   89
           4   9442   8641
           5   87868162   80946721
           6   7625001202225282   7064733248680321
           7   57512842974398078691295382996482
53554699334818911197308393584001
           8  3.276393304854688402032060470960911E+0063
              3.064416112313000256454554621335885E+0063
  M =            9
           2   11   10
           3   119   109
           4   13961   12871
           5   192533321   178503931
           6   36675432127744841   34171130863212751
           7  1.332544272109744425180555122579081E+0033
              1.246969465878509204729330985899491E+0033
           8  1.761028142209461685699744021414489E+0066
              1.654318971790641381597454949851900E+0066
  M =           10
           2   12   11
           3   142   131
           4   19922   18481
           5   392733122   366102001
           6   152820871904433602   143071165217460481
           7  2.316410532866728667414098247373312E+0034
              2.176920343243357602322692614700192E+0034
           8  5.326842730773593818395794488418642E+0068
              5.023183699299603840466455092616320E+0068

Regards

Hugo