[seqfan] Fathers and ghosts

[Eric Angelini]

Hello Seqfans,
Let's pick any positive integer and call it the "father" -- for instance 148.
Lets call 347 the "ghost" produced by 148 according to this first rule:

1 4 8
 3 4 7

3 comes from |1-4|
4 comes from |4-8|
7 comes from |8-1|

We produce now a new father f' with the second rule:
--- if f<g then f'=f+g
--- if f>g then f'=f-g

Here we compute f' doing 148+347 as 148<347 --- thus f'=495.

In iterating this procedure, it seems that all integers enter into a loop.

Our starting example, for instance, will produce this sequence of
operations:

       1 4 8
   +  3 4 7
   ----------
       4 9 5
   +  5 4 1
   ----------
     1 0 3 6
 +  1 3 3 5
   -----------
     2 3 7 1
 -   1 4 6 1
   -----------
        9 1 0
    -   8 1 9
       -------
           9 1
       -   8 8
          -----
              3
          -   0
             ---
              3    <---- loop

We could compute S, S being the sequence of all integers which are 
part of a loop (like 3, seen here);

S starts, I guess, with 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,...
but 16 is not in S as 16 enters into the "5" loop like this:

    1 6
+  5 5
   -----
    7 1
 -  6 6
   -----
      5
   -  0
     ---
      5

The integer 17 is not in S either, as 17 produces this succession of 
fathers before entering into a loop:

17--83--28--94--39--105--259--596--165--679--556--545--435--314--83 
(loop, see 2nd and last term)

All the above fathers are part of S, of course -- but not 17.

Two remaks:
1) when a ghost starts with a leading zero, we erase this zero
immediately -- example:

    1 1 7
 -     6 6
   --------
       5 1

2) is it possible that some integers never end in a loop?

Have a great 2 0 1 4 !
              +   2 1 3 2
                  -----------
                   4 1 4 6
              -    3 3 2 2
                  -----------
                      8 2 4
                 -    6 2 4
                     --------
                      2 0 0
                 +   2 0 2
                      -------
                      4 0 2
                  +  4 2 2
                      -------
                      8 2 4  <--  loop!

Best,
É.