[seqfan] about a certain formula for A063003

[Simon Plouffe]

Yes, you are right about the formula

-x*(-1-4*x+16*x^2-32*x^3+128*x^4-256*x^5-1024*x^6+4096*x^7-8192*x^8+
32768*x^9-65536*x^10-262144*x^11+1572864*x^12)/(3*x-1)/(524288*x^12-1)


  but you know sometimes even an error can lead
to something interesting,

  The formula found, matches the first 52 terms,

  [1, 7, 5, 47, 13, 295, 1909, 1631, 13085, 6487, 84997, 517135, 502829, 
3605639,

     2428309, 24062143, 5077565, 149450423, 985222181, 808182895, 
6719515981,

     2978678759, 43295774645, 267326277407, 252223018333, 1856180682775,

     1170495537221, 12307579633871, 1738366812781, 75583844616007,

     508226510558677, 398779624833407, 3448138688185469, 1337216809815415,

     22026048938928229, 138135740854712623, 126176846412426125,

     954991291540701863, 559130865408411637, 6289078614652622815,

     420491770248316829, 38154963458164053719, 262038842964168574085,

     195820718533800070543, 1768053776318811515053]

 > guessgf(%,x);
                    2       3        4        5         6         
7         8
[- (-1 - 4 x + 16 x  - 32 x  + 128 x  - 256 x  - 1024 x  + 4096 x  - 8192 x

               9          10           11            12    /
      + 32768 x  - 65536 x   - 262144 x   + 1572864 x  )  /  (
                                                         /

                        13           12
     1 - 3 x + 1572864 x   - 524288 x  ), ogf]

 > factor(%);
                    2       3        4        5         6         
7         8
[- (-1 - 4 x + 16 x  - 32 x  + 128 x  - 256 x  - 1024 x  + 4096 x  - 8192 x

               9          10           11            12    /
      + 32768 x  - 65536 x   - 262144 x   + 1572864 x  )  /  ((3 x - 1)
                                                         /

              12
     (524288 x   - 1)), ogf]

  Now if we make the difference between that formula
and the EXACT expression in the sequence (taking the first 999 terms)

  We find :

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0,

     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0,

     0, -19342813113834066795298816, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

     -10141204801825835211973625643008, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

     -5316911983139663491615228241121378304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0,

     -2787593149816327892691964784081045188247552, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0]

these numbers are 2^k and k is : [84, 103, 122, 141],
the first difference is 19, 19 , 19,

  This is quite interesting, isn't ?

It goes like that for a couple of hundreds terms like this : (factored),

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, -
``(2)^84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -``(2)^103, 0, 0, 0, 0, 0, 0, 
0, 0,
0, 0, 0, -``(2)^122, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -``(2)^141, 0, 0, 
0, 0, 0
, 0, 0, 0, 0, 0, 0, -``(2)^160, 0, 0, 0, 0, -``(2)^168, 0, 0, 0, 0, 0, 
0, -``(2
)^179, 0, 0, 0, 0, -``(2)^187, 0, 0, 0, 0, 0, 0, -``(2)^198, 0, 0, 0, 0, 
-``(2)
^206, 0, 0, 0, 0, 0, 0, -``(2)^217, 0, 0, 0, 0, -``(2)^225, 0, 0, 0, 0, 
0, 0, -
``(2)^236, 0, 0, 0, 0, -``(2)^244, 0, 0, 0, 0, -``(2)^252, 0, 
-``(2)^255, 0, 0,
0, 0, -``(2)^263, 0, 0, 0, 0, -``(2)^271, 0, -``(2)^274, 0, 0, 0, 0, 
-``(2)^282
, 0, 0, 0, 0, -``(2)^290, 0, -``(2)^293, 0, 0, 0, 0, -``(2)^301, 0, 0, 
0, 0, -
``(2)^309, 0, -``(2)^312, 0, 0, 0]

and then the pattern breaks,

I wonder now if there could be another type of formula
knowing this kind of behavior, I am not certain but
it might just be.

  So, what I am saying is this : despite the fact that
the formula is inexact, it leads to an interesting
question anyway.

  Bonne journée,
  Simon Plouffe