A357049 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.

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

Offset

0,2

Links

Table of n, a(n) for n=0..69.

Eric Angelini, Un tableau jaune de lui-même, personal blog CinquanteSignes.blogspot.com, Jun. 2, 2022.

Example

If a square array is filled along falling diagonals with the digits of terms, we get:
  [ 0 2 6 3 0 2 0 ...]
  [ 4 1 2 5 7 1 8 ...]
  [ 8 1 2 9 2 2 2 ...]
  [ 2 3 2 1 6 7 3 ...]
  [ 2 4 4 1 2 2 8 ...]
  [ 1 1 1 9 1 1 1 ...]
  [ 6 3 4 3 5 4 5 ...]
  [ 0 1 4 5 5 4 5 ...]
  [ 2 0 0 5 3 6 3 ...]
  [ 3 1 2 5 5 6 7 ...]
  [ 7 3 6 1 5 8 1 ...]
  [ ...   ...     ...]
If the odd digits are replaced by blanks, we get:
  [ 0 2 6 _ 0 2 0 ...]
  [ 4 _ 2 _ _ _ 8 ...]
  [ 8 _ 2 _ 2 2 2 ...]
  [ 2 _ 2 _ 6 _ _ ...]
  [ 2 4 4 _ 2 2 8 ...]
  [ _ _ _ _ _ _ _ ...]
  [ 6 _ 4 _ _ 4 _ ...]
  [ 0 _ 4 _ _ 4 _ ...]
  [ 2 0 0 _ _ 6 _ ...]
  [ _ _ 2 _ _ 6 _ ...]
  [ _ _ 6 _ _ 8 _ ...]
This reproduces the first row "0 2 ..." and second row "4 1 ..." of the square array.
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.

Prog

(PARI)
{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]]}
L357049/*terms*/= D357049/*digits*/= List(0); /*bitmap of used terms*/U357049=1
{parity(n, x=divrem(A025581(n), 4), y=divrem(A002262(n), 6))=
  if(x[2] == 3 || y[2]==5, 1/* space between rows/col's: digit must be odd */,
  !digit[D357049[binomial(x[1]+y[1]+1, 2) +y[1] +1]+1][y[2]+1, x[2]+1])}
A357049(n)={while(#L357049<=n, /* extend the list by a new term */
  my(k=valuation(U357049+1, 2), d); until(!k++, bittest(U357049, k) ||
  for(i=1, #d=digits(k), d[i]%2 == parity(i-1+#D357049) || next(2)) || break);
  listput(L357049, k); for(i=1, #d, listput(D357049, d[i])); U357049 += 1<<k);
  L357049[n+1]} /* end of A357049 */

Crossrefs

Sequence in context: A080413 A004517 A254350 * A364864 A056649 A376556

Adjacent sequences: A357046 A357047 A357048 * A357050 A357051 A357052

Keyword

nonn,base

Author

Eric Angelini and M. F. Hasler, Oct 20 2022

Status

approved