Sequence & Its Numerators & Denominators

[Pfoertner, Hugo]

Leroy, SeqFans,

a few results (what I can do without BigNums):

  M = 1
  k                                  n                                  d
  2                                  3                                  2
  3                                 21                                 10
  4                                861                                310
  5                            1275141                             363010
  6                      2551762438701                       594665194510
  7          9546380157472159016030421          1871071000515058250871610

  M = 2
  2                                  4                                  3
  3                                 56                                 33
  4                               9968                               4785
  5                          294115808                          118289985
  6                 242590126064151488                  83574429584465985
  7  1.582486013449121321571784280714996E+0035
4.753334809763917319511326686829158E+0034

  M = 3
  2                                  5                                  4
  3                                115                                 76
  4                              57155                              31996
  5                        13457544835                         6509938156
  6              718532108172999980195              305202648617286353116
  7  1.987460976488531436231264449305835E+0042
7.510423643158930656025467393343864E+0041

  M = 4
  2                                  6                                  5
  3                                204                                145
  4                             225624                             139345
  5                       266503910064                       145175334145
  6           361476924706582858606944           175835054422073316222145
  7  6.497828978375256831345904988205142E+0047
2.851592252760362666671714028680662E+0047

  M = 5
  2                                  7                                  6
  3                                329                                246
  4                             703073                             465186
  5                      3125677649801                      1851696597486
  6         60424917190276611506698577         32367813634013973326521626
  7  2.216749800419013098487257188463868E+0052
1.082678765177426393429230538456141E+0052

  M = 6
  2                                  8                                  7
  3                                496                                385
  4                            1858016                            1293985
  5                     25521830405056                     16099866182785
  6       4729979071868915640350500736       2724594016670394605535321985
  7  1.600107574782880396587666104603987E+0056
8.474704862481751818275190171802905E+0055


  M = 7
  2                                  9                                  8
  3                                711                                568
  4                            4346343                            3149560
  5                    159613018433703                    105743204607160
  6     212090193707026209909082272423     129327569784001903386257059960
  7  3.697339705210831279240508910851950E+0059
2.087293855962712766461507748054839E+0059

  M = 8
  2                                 10                                  9
  3                                980                                801
  4                            9253160                            6921441
  5                    813058161891920                    560267953544961
  6    6199569461904967962915175521440    3958143639579167493986989412481
  7  3.565548649714140138596310964638573E+0062
2.119771925416881455775058496250262E+0062

  M = 9
  2                                 11                                 10
  3                               1309                               1090
  4                           18274949                           14029390
  5                   3518536622075629                   2504301264532090
  6  129043851071919331586456316579989   85574806231235220451224136679590
  7  1.719566445968690086105739864765900E+0065
1.067091704188203041022890715625893E+0065

  M =10
  2                                 12                                 11
  3                               1704                               1441
  4                           33947088                           26631121
  5                  13332145853048736                   9749706686973121
  6 2037430153619986537231286982027072 1394901896233710650914056326731201
  7  4.719524667825613862537115696813998E+0067
3.036590314739899779293393958023222E+0067

Fortran program:

C a[k+1] = a[k] + 1 / ( m + 1 / a[k] ), a[1] = 1
C Proposed by Leroy Quet
C Hugo Pfoertner, 26.02.2003
C
      REAL*16 A, M, ONE, N, D, NN, DD, MXINT
      PARAMETER ( ONE = 1.0Q0, MXINT=1.0Q34 )
      INTEGER K, MM, I
      DO 10 MM = 1, 10
      WRITE (*,*) ' '
      WRITE (*,*) ' M = ', MM
      WRITE (10,*) ' M = ', MM
      M = QEXT (MM)
      A = ONE
      N = ONE
      D = ONE
      DO 20 I = 2, 7
      A = A + ONE / ( M + ONE / A )
      NN = N * ( M*N + D + D )
      DD = D * ( M*N + D )
      WRITE (*,*) I, A, A - QEXT(NN)/QEXT(DD)
      N = NN
      D = DD
      IF ( N .LE. MXINT  .AND. D .LE. MXINT ) THEN
        WRITE (10,1000) I, N, D
1000    FORMAT ( I3, 2F36.0 )
      ELSE
        WRITE (10,*) I, N, D
      ENDIF
20    CONTINUE
10    CONTINUE
      END

Currently I am too busy to submit the sequences to the OEIS.

Best Regards

Hugo Pfoertner

-----Ursprüngliche Nachricht-----
Von: Leroy Quet [mailto:qqquet at mindspring.com]
Gesendet am: 26 February, 2003 04:10
An: seqfan at ext.jussieu.fr
Betreff: Sequence & Its Numerators & Denominators

I just submitted this to sci.math. I already sent the m=1 example to 
seq.fan a little while back. Perhaps someone could calculate the 
numerator/denominator sequences for the first couple of m's and post them 
to the EIS.

Thanks,
Leroy Quet

---------------


This is a generalization of the sequence discussed at:
http://mathforum.org/discuss/sci.math/t/482979

Given a fixed positive integer m;

Let a[1] = any nonzero rational where its denominator is relatively
prime to m.

Let:
                     1
a[k+1] = a[k] + -------------  , for all positive integers k.
                 m + 1 /a[k]

In linear mode:

a[k+1] = a[k] + 1/(m + 1 /a[k]) , for all positive integers k.

If n[k] = numerator of a[k] in reduced form,

and d[k] = denominator of a[k] in reduced form,

then, for k = all positive integers,


n[k+2] =

 n[k+1]*(m*n[k+1] + n[k+1]^2/(2n[k]^2) -m^2 *n[k]^2/2),
 

and d[k+2] = 

 d[k+1]*(d[k+1] + d[k+1]^2/d[k]^2 -d[k]^2)



The above are from these recursions:

n[k+1] = n[k] *(m*n[k] + 2*d[k]),

d[k+1] = d[k] *(m*n[k] + d[k])


Noteworthy facts:

*d[2] involves m. But the recursion, for d[k+2] in terms of d[k] and
d[k+1], itself is independent of m.

*Each d[k+1] is divisible by d[k], and each n[k+1] is divisible by
n[k].

*(d[k+1]/d[k]) is relatively prime to (n[j+1]/n[j]) for every positive
integer k and every positive integer j.

Thanks,
Leroy Quet