A262620 - OEIS

A262620

Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton on the square grid (see Comments lines for definition).

1, 5, 17, 21, 49, 69, 81, 85, 145, 197, 241, 277, 305, 325, 337, 341, 465, 581, 689, 789, 881, 965, 1041, 1109, 1169, 1221, 1265, 1301, 1329, 1349, 1361, 1365, 1617, 1861, 2097, 2325, 2545, 2757, 2961, 3157, 3345, 3525, 3697, 3861, 4017, 4165, 4305, 4437, 4561, 4677, 4785, 4885, 4977, 5061, 5137, 5205, 5265, 5317, 5361, 5397

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OFFSET

0,2

COMMENTS

On the infinite square grid consider four 90-degree wedges forming a "X" with the vertex located at the center of a cell.

At stage 0 we start with an ON cell in the vertex of the wedges, so a(0) = 1.

In order to construct the structure we use the following rules for the South wedge:

- The cells turned ON remain ON forever.

- At stage 1 we turn ON the nearest cell to the initial ON cell.

- If n is a power of 2, at stage n we turn "ON" 2*n - 1 connected cells in the n-th row of the wedge.

- Otherwise, if n is not a power of 2, at stage n we turn "ON" k - 2 connected cells in the n-th row of the wedge, where k is the number of ON cells in row n - 1.

- The "ON" cells of row n must be centered respect to the "ON" cells of row n - 1.

The structures in the other three wedges are copies of the structure in the South wedge but they grow in direction East, North and West.

Note that in every wedge the structure seems to grow into the holes of a virtual structure similar to the Sierpiński's triangle but using square cells.

A262621 gives the number of cells turned "ON" at n-th stage.

This is analog of A256266, but here we are working on the square grid and we have four wedges, not six wedges.

LINKS

Table of n, a(n) for n=0..59.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to cellular automata

FORMULA

a(n) = 1 + 4*A261692(n).

EXAMPLE

Illustration of the structure after 15 generations:

. O

. O O O

. O O O O O

. O O O O O O O

. O O O O O O O O O

. O O O O O O O O O O O

. O O O O O O O O O O O O O

. O O O O O O O O O O O O O O O

. O O O

. O O O O O O O

. O O O O O O O O O O O

. O O O O O O O O O O O O O O O

. O O O O O O O O O O O O O

. O O O O O O O O O O O O O O O O O O O

. O O O O O O O O O O O O O O O O O O O O O O O

. O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

. O O O O O O O O O O O O O O O O O O O O O O O

. O O O O O O O O O O O O O O O O O O O

. O O O O O O O O O O O O O

. O O O O O O O O O O O O O O O

. O O O O O O O O O O O

. O O O O O O O

. O O O

. O O O O O O O O O O O O O O O

. O O O O O O O O O O O O O

. O O O O O O O O O O O

. O O O O O O O O O

. O O O O O O O

. O O O O O

. O O O

. O

There are 341 ON cells in the structure, so a(15) = 341.

Note that every circle in the structure should be replaced with a square cell.

CROSSREFS

Cf. A147562, A160720, A256266, A261692, A261693, A262621.

Sequence in context: A224513 A302423 A305478 * A191210 A191143 A032376

Adjacent sequences: A262617 A262618 A262619 * A262621 A262622 A262623

KEYWORD

nonn,look

AUTHOR

Omar E. Pol, Oct 16 2015

STATUS

approved