Un tableau jaune de lui-même

A yellow array of itself

We want to produce an array A of digits that embeds a copy of A, this copy embedding itself another copy of A, and so on, recursively.

As the array is infinite, we fill A with digits by its successive antidiagonals; hereunder are the 34 first complete such antidiagonals (we will explain later why the digits forming A were chosen, and not others):

If we underline in yellow the even (black) digits of A, we get this:

We see now (above) that the yellow "big digits" reproduce the array A (the first horizontal yellow digits are 0, 2, 6, 3, 0, 2, 0 and 1 – which are also the black (small) digits of the upper row of A.

This is the beginning of the recursion.

Indeed (look now hereunder), if we put every even "big yellow digit" in a (blue) box, the said boxes will form again the array A (note that the fourth "big yellow digit" of the upper row is "3", and as "3" is not even, we don't draw a blue box around it – neither do we around the "big yellow digits 1, 5, 7 and 9, of course):

How were chosen the black digits forming the "ground level" of the array?

We decided that those digits would come, one after the other, from a sequence S of integers. And we decided that those integers would be distinct. And that the sequence S would be the lexicographically earliest of its kind.

Here is the start of the filling procedure; we draw (with a pencil) a little zero in the upper left blue angle of the array A:

This 0 is immediately surrounded by another zero that we draw – that new zero having a rectangular shape (we will use the 3x5 "rectangular font" to draw the other digits, as explained here):

The red line below shows the 2nd antidiagonal of A (the 1st one is occupied by the "small pencil zero"):

Antidiagonals are usually filled starting from the top, then along successive (red) lines towards S.-W. (or down/left).

We've arbitrarily decided to fill the body of a "rectangular digit" with small even digits; the inside "holes" of a rectangular digit will be occupied by small odd digits; the spaces between big rectangular digits will also be filled with little odd digits. We then get the hereunder first fillings:

Explanation:

After 0, the smallest even digits not yet used to fill the "body" of the rectangular zero are 2 and 4; we use them to complete the second antidiagonal of A.

The next antidiagonal (the third one) will be formed by 6, 1 and 8, in that order. The digit "1" is indeed NOT belonging to the body of the rectangular zero ("1" is placed in a "hole") and as "1" is the smallest unused odd digit so far, we put "1" between 6 and 8 on the third antidiagonal.

The next operation we must explain is the "expansion" that appears with the other "big rectangular digits" in the upper left angle of the array.

You will notice immediately that those big rectangular digits are the same as the small pencil digits we've used before.

This is the key of the recursion: when you add a "small pencil digit" to the array, you must add accordingly the same rectangular digit (properly aligned – the rectangular 1x5 "1s" are centered)) on the grid (along a "second level" antidiagonal!)

The last thing to explain is the sequence S that slowly appears on top of the page: for now S is composed (see above) of 0, 2, 4, 6, 1, 8,... (It will be this S that will be submitted to the OEIS.) And this S has to be the lexicographically earliest made of distinct nonnegative terms not leading to a contradiction.

We go on filling A by its fourth antidiagonal (and expanding the figure with the according "big rectangular digits" we need):

We see above how to deal with the numbers having more than one digit that will appear in S: we just turn those numbers in a succession of digits – and fill the array with them, one by one as they "arrive"; this explains why the 4th antidiagonal (in red, above) starts with 3 and goes on with 21 

This 4th antidiagonal will end with the digits "2" of 20 (see hereunder, in red):

The fifth antidiagonal starts with the "0" of 20 we left behind, then goes on with 5, 23 and the starting "2" of 22 (the second "2" of 22 will start the sixth antidiagonal (this is also true for the "expanded" array made of "big rectangular digits", see hereunder):

The sixth antidiagonal is completed hereunder, both in the "pencil small digits" world and in the "big rectangular digits" world; the sequence S goes now (on top of the hereunder page) 0, 2, 4, 6, 1, 8, 3, 21, 20, 5, 23, 22, 7, 9, 24, 10, ...

As always, we advance simultaneously on three fronts: 

a) extend S with the necessary "small pencil digits", depending of the odd/even criterion fixed by the "bodies" of the "big rectangular digits";

b) extend the array of the "big rectangular digits";

c) extend S on the top of the page [no duplicates, smallest necessary (unused) terms first].

The seventh antidiagonal goes like this:

The sequence S so far (top of the page, below), together with the 8th and 9th antidiagonal:

A few more antidiagonals to fill... and my page has no more room:

For the records, hereunder is S, computed by hand as far as possible:

S =

0,2,4,6,1,8,3,21,20,5,23,22,7,9,24,10,12,14,16,18,26,11,30,25,27,29,41,28,43,32,13,40,34,36,38,15,50,17,19,42,31,45,52,33,47,49,61,54,35,63,44,37,46,39,56,51,58,48,70,72,74,53,65,76,78,90,67,55,69,57,81,92,60,83,85,87,89,211,59,94,96,62,98,213,64,215,217,66,101,210,103,212,71,219,214,105,231,68,73,233,107,75,216,109,218,121,230,77,235,232,237,80,79,123,91,239,125,100,82,127,234,129,236,141,93,238,143,84,251,253,86,95,88,255,257,145,250,147,259,149,252,161,254,163,200,97,271,273,99,203,165,102,111,256,167,275,169,110,181,258,183,270,185,272,112,114,113,205,116,118,277,202,130,187,189,301,274,303,276,305,278,104,132,279,307,134,136,291,204,115,293,295,297,290,138,150,309,299,321,206,323,411,152,117,201,325,106,413,415,327,108,119,292,154,156,329,158,341,417,419,294,343,131,296,133,431,208,170,135,433,345,137,298,139,347,...

S forms T (the successive digits of S that were used to fill A by its antidiagonals):

T =

0,2,4,6,1,8,3,2,1,2,0,5,2,3,2,2,7,9,2,4,1,0,1,2,1,4,1,6,1,8,2,6,1,1,3,0,2,5,2,7,2,9,4,1,2,8,4,3,3,2,1,3,4,0,3,4,3,6,3,8,1,5,5,0,1,7,1,9,4,2,3,1,4,5,5,2,3,3,4,7,4,9,6,1,5,4,3,5,6,3,4,4,3,7,4,6,3,9,5,6,5,1,5,8,4,8,7,0,7,2,7,4,5,3,6,5,7,6,7,8,9,0,6,7,5,5,6,9,5,7,8,1,9,2,6,0,8,3,8,5,8,7,8,9,2,1,1,5,9,9,4,9,6,6,2,9,8,2,1,3,6,4,2,1,5,2,1,7,6,6,1,0,1,2,1,0,1,0,3,2,1,2,7,1,2,1,9,2,1,4,1,0,5,2,3,1,6,8,7,3,2,3,3,1,0,7,7,5,2,1,6,1,0,9,2,1,8,1,2,1,2,3,0,7,7,2,3,5,2,3,2,2,3,7,8,0,7,9,1,2,3,9,1,2,3,9,1,2,5,1,0,0,8,2,1,2,7,2,3,4,1,2,9,2,3,6,1,4,1,9,3,2,3,8,1,4,3,8,4,2,5,1,2,5,3,8,6,9,5,8,8,2,5,5,2,5,7,1,4,5,2,5,0,1,4,7,2,5,9,1,4,9,2,5,2,1,6,1,2,5,4,1,6,3,2,0,0,9,7,2,7,1,2,7,3,9,9,2,0,3,1,6,5,1,0,2,1,1,1,2,5,6,1,6,7,2,7,5,1,6,9,1,1,0,1,8,1,2,5,8,1,8,3,2,7,0,1,8,5,2,7,2,1,1,2,1,1,4,1,1,3,2,0,5,1,1,6,1,1,8,2,7,7,2,0,2,1,3,0,1,8,7,1,8,9,3,0,1,2,7,4,3,0,3,2,7,6,3,0,5,2,7,8,1,0,4,1,3,2,2,7,9,3,0,7,1,3,4,1,3,6,2,9,1,2,0,4,1,1,5,2,9,3,2,9,5,2,9,7,2,9,0,1,3,8,1,5,0,3,0,9,2,9,9,3,2,1,2,0,6,3,2,3,4,1,1,1,5,2,1,1,7,2,0,1,3,2,5,1,0,6,4,1,3,4,1,5,3,2,7,1,0,8,1,1,9,2,9,2,1,5,4,1,5,6,3,2,9,1,5,8,3,4,1,4,1,7,4,1,9,2,9,4,3,4,3,1,3,1,2,9,6,1,3,3,4,3,1,2,0,8,1,7,0,1,3,5,4,3,3,3,4,5,1,3,7,2,9,8,1,3,9,3,4,7,...

____________________