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

Eric Angelini Eric.Angelini at kntv.be
Sun Oct 13 00:24:29 CEST 2013

```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

```

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