A357049 - OEIS

%I #18 Oct 23 2022 01:45:58

%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,

%T 41,28,43,32,13,40,34,36,38,15,50,17,19,42,31,45,52,33,47,49,61,54,35,

%U 63,44,37,46,39,56,51,58,48,70,72,74,53,65,76,78,90,67,55,69,57

%N Lexicographically earliest sequence of distinct nonnegative integers such that, when the digits fill a square array read by falling antidiagonals, the "bitmap" of even digits reproduces the same square array.

%H Eric Angelini, <a href="https://cinquantesignes.blogspot.com/2022/06/un-tableau-jaune-de-lui-meme.html">Un tableau jaune de lui-même</a>, personal blog CinquanteSignes.blogspot.com, Jun. 2, 2022.

%e If a square array is filled along falling diagonals with the digits of terms, we get:

%e   [ 0 2 6 3 0 2 0 ...]

%e   [ 4 1 2 5 7 1 8 ...]

%e   [ 8 1 2 9 2 2 2 ...]

%e   [ 2 3 2 1 6 7 3 ...]

%e   [ 2 4 4 1 2 2 8 ...]

%e   [ 1 1 1 9 1 1 1 ...]

%e   [ 6 3 4 3 5 4 5 ...]

%e   [ 0 1 4 5 5 4 5 ...]

%e   [ 2 0 0 5 3 6 3 ...]

%e   [ 3 1 2 5 5 6 7 ...]

%e   [ 7 3 6 1 5 8 1 ...]

%e   [ ...   ...     ...]

%e If the odd digits are replaced by blanks, we get:

%e   [ 0 2 6 _ 0 2 0 ...]

%e   [ 4 _ 2 _ _ _ 8 ...]

%e   [ 8 _ 2 _ 2 2 2 ...]

%e   [ 2 _ 2 _ 6 _ _ ...]

%e   [ 2 4 4 _ 2 2 8 ...]

%e   [ _ _ _ _ _ _ _ ...]

%e   [ 6 _ 4 _ _ 4 _ ...]

%e   [ 0 _ 4 _ _ 4 _ ...]

%e   [ 2 0 0 _ _ 6 _ ...]

%e   [ _ _ 2 _ _ 6 _ ...]

%e   [ _ _ 6 _ _ 8 _ ...]

%e This reproduces the first row "0 2 ..." and second row "4 1 ..." of the square array.

%e Since this gives back the original array, the process can be iterated infinitely: e.g., removing the odd digits (made of "elementary" even digits: e.g., the "1" made of 4-4-6-6-8) will again yield the same array, with the entries written in even larger digits whose pixels are the even digits made of even smaller even digits. And so on.

%o (PARI)

%o {digit=[[1,1,1; 1,0,1; 1,0,1; 1,0,1; 1,1,1], [0,1,0; 0,1,0; 0,1,0; 0,1,0; 0,1,0],[1,1,1; 0,0,1; 1,1,1; 1,0,0; 1,1,1], [1,1,1; 0,0,1; 1,1,1; 0,0,1; 1,1,1], [1,0,1; 1,0,1; 1,1,1; 0,0,1; 0,0,1], [1,1,1; 1,0,0; 1,1,1; 0,0,1; 1,1,1], [1,1,1; 1,0,0; 1,1,1; 1,0,1; 1,1,1], [1,1,1; 0,0,1; 0,0,1; 0,0,1; 0,0,1], [1,1,1; 1,0,1; 1,1,1; 1,0,1; 1,1,1], [1,1,1; 1,0,1; 1,1,1; 0,0,1; 1,1,1]]}

%o L357049/*terms*/= D357049/*digits*/= List(0); /*bitmap of used terms*/U357049=1

%o {parity(n, x=divrem(A025581(n), 4), y=divrem(A002262(n), 6))=

%o   if(x[2] == 3 || y[2]==5, 1/* space between rows/col's: digit must be odd */,

%o   !digit[D357049[binomial(x[1]+y[1]+1, 2) +y[1] +1]+1][y[2]+1, x[2]+1])}

%o A357049(n)={while(#L357049<=n, /* extend the list by a new term */

%o   my(k=valuation(U357049+1,2), d); until(!k++, bittest(U357049,k) ||

%o   for(i=1, #d=digits(k), d[i]%2 == parity(i-1+#D357049) || next(2)) || break);

%o   listput(L357049, k); for(i=1, #d, listput(D357049, d[i])); U357049 += 1<<k);

%o   L357049[n+1]} /* end of A357049 */

%K nonn,base

%O 0,2

%A _Eric Angelini_ and _M. F. Hasler_, Oct 20 2022