[seqfan] A fractal seq with an "even-sum eraser"

[Eric Angelini]

Hello SeqFans,
Read this binary seq "B" with an eraser -- and erase two consecutive
terms of "B" when their sum is even. Then continue the reading/erasing
procedure from there:

Start:
B = 1,0,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,
    0,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,
    0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,...

Erasures of 0,0, are marked with a dot, erasures of 1,1, with a star:

B = 1,. . * * 0,* * 0,1,. . 1,0,1,. . 1,0,1,0,* * 0,1,0,1,0,* * 0,1,0,1,0,1,
    . . 1,0,1,0,1,0,1,. . 1,0,1,0,1,0,1,0,* * 0,1,0,1,0,1,0,1,0,* * 0,1,0,1,
    0,1,0,1,0,1,. . 1,0,1,0,1,0,1,0,1,0,1,0,...

Without dots and stars we see that this seq reproduces the original "B" one:

B = 1,        0,    0,1,    1,0,1,    1,0,1,0,    0,1,0,1,0,    0,1,0,1,0,1,
        1,0,1,0,1,0,1,    1,0,1,0,1,0,1,0,    0,1,0,1,0,1,0,1,0,    0,1,0,1,
    0,1,0,1,0,1,    1,0,1,0,1,0,1,0,1,0,1,0,...

This last "B" presentation can be seen as an array (without the initial "1"):

0
0,1,
1,0,1,
1,0,1,0,
0,1,0,1,0,
0,1,0,1,0,1,
1,0,1,0,1,0,1,
1,0,1,0,1,0,1,0,
0,1,0,1,0,1,0,1,0,
0,1,0,1,0,1,0,1,0,1,
1,0,1,0,1,0,1,0,1,0,1,
1,0,1,0,1,0,1,0,1,0,1,0,  ...

  or:
                      0
                    0   1
                  1   0   1
                1   0   1   0
              0   1   0   1   0
            0   1   0   1   0   1
          1   0   1   0   1   0   1
        1   0   1   0   1   0   1   0
      0   1   0   1   0   1   0   1   0
    0   1   0   1   0   1   0   1   0   1
  1   0   1   0   1   0   1   0   1   0   1
1   0   1   0   1   0   1   0   1   0   1   0   ...

... where the "fractality" of the seq is obvious; erase the triangle's
two oblique sides, and you'll find the initial triangle:

                      .
                    .   *
                  *   0   *
                *   0   1   .
              .   1   0   1   .
            .   1   0   1   0   *
          *   0   1   0   1   0   *
        *   0   1   0   1   0   1   .
      .   1   0   1   0   1   0   1   .
    .   1   0   1   0   1   0   1   0   *
  *   0   1   0   1   0   1   0   1   0   *
*   0   1   0   1   0   1   0   1   0   1   .   ...

The binary seq "B" can be easely transformed in a decimal one, where all
integers will appear at least once (we start "D" here with 1,0):

D = 1,0, then:

2,
0,3,
5,2,7,
9,0,3,4,
6,5,2,7,8,
10,9,0,3,4,11,
13,6,5,2,7,8,15,
17,10,9,0,3,4,11,12,
14,13,6,5,2,7,8,15,16,
18,17,10,9,0,3,4,11,12,19,
21,14,13,6,5,2,7,8,15,16,23,
25,18,17,10,9,0,3,4,11,12,19,20,  ...

   or:

D = 1,0, then:
                               2
                             0    3
                          5    2    7
                        9    0    3    4
                     6    5    2    7    8
                  10    9    0    3    4    11
               13    6    5    2    7    8    15
            17    10    9    0    3    4    11    12
         14    13    6    5    2    7    8    15    16
      18    17    10    9    0    3    4    11    12    19
   21    14    13    6    5    2    7    8    15    16    23
25    18    17    10    9    0    3    4    11    12    19    20   ...

Again, the "fractality" is obvious when you erase the two oblique sides.

Would "D" be of interest for the OEIS?

> Start reading "D" and erase two consecutive terms of "D" when their sum
  is even. Then continue the reading-erasing process from there. The result
  will reproduce "D":

D = 1,0,2,0,3,5,2,7,9,0,3,4,6,5,2,7,8,10,9,0,3,4,11,13,6,5,2,7,8,15,17,10,
    9,0,3,4,11,12,14,13,6,5,2,7,8,15,16,18,17,10,9,0,3,4,11,12,19,21,14,13,
    6,5,2,7,8,15,16,23,25,18,17,10,9,0,3,4,11,12,19,20,...

Best,
É.
_____________________________________________________________________________________

P.-S.

I haven't yet computed the fractal seq with an "odd-sum eraser"

And yes, this has a 2-year ago "prime-sum eraser" flavor (not in the OEIS):
http://www.cetteadressecomportecinquantesignes.com/ErasePrimeSums.htm