A393834 A variant of Sierpinski carpet, read by antidiagonals.

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

0

Comments

000 111

Fixed point of the morphism 0 -> 010, 1 -> 101, starting from A(0, 0) = 1.

000 111

This array is 3-automatic, self-similar and nonperiodic.

The average value after n iterations equals A081190(n)/A001019(n) and tends to 1/2.

A(n, k) = 1 when the ternary expansions of n and k both equals 1 at an even number of positions, A(n, k) = 0 otherwise.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10010

Rémy Sigrist, Representation of the sixth iteration (black pixels denote 1's, white pixels denote 0's)

Index entries for sequences related to automatic sequences

Formula

A(n, k) >= A153490(n+1, k+1).

A(n, k) = A(k, n).

A(n, n) = (1+n) mod 2.

A(n, 2*n) = 1.

A(3*n, 3*k) = A(n, k).

Example

The array begins:
  n\k | 0  1  2  3  4  5  6  7  8
  ----+--------------------------
    0 | 1  1  1  1  1  1  1  1  1
    1 | 1  0  1  1  0  1  1  0  1
    2 | 1  1  1  1  1  1  1  1  1
    3 | 1  1  1  0  0  0  1  1  1
    4 | 1  0  1  0  1  0  1  0  1
    5 | 1  1  1  0  0  0  1  1  1
    6 | 1  1  1  1  1  1  1  1  1
    7 | 1  0  1  1  0  1  1  0  1
    8 | 1  1  1  1  1  1  1  1  1

Mathematica

A393834list[i_] := Table[Diagonal[#, d], {d, 1 - 3^i, 0}] & [Nest[ArrayFlatten[{{#, #, #}, {#, 1 - #, #}, {#, #, #}}] &, {{1}}, i]]; (* Generates 3^i antidiagonals *)
A393834list[3] (* Paolo Xausa, Apr 05 2026 *)

Prog

(PARI) A(x, y) = { my (v = 1); while (x || y, if (x%3==1 && y%3==1, v = 1-v; ); x \= 3; y \= 3; ); return (v); }

Crossrefs

Cf. A001019, A081190, A153490.

Sequence in context: A071026 A259022 A153490 * A394841 A014194 A014379

Adjacent sequences: A393831 A393832 A393833 * A393835 A393836 A393837

Keyword

nonn,tabl,base

Author

Rémy Sigrist, Apr 03 2026

Status

approved