A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

0, 3, 12, 21, 48, 57, 84, 111, 192, 201, 228, 255, 336, 363, 444, 525, 768, 777, 804, 831, 912, 939, 1020, 1101, 1344, 1371, 1452, 1533, 1776, 1857, 2100, 2343, 3072, 3081, 3108, 3135, 3216, 3243, 3324, 3405, 3648, 3675, 3756, 3837, 4080, 4161, 4404, 4647

Offset

0,2

Comments

From Omar E. Pol, Nov 10 2009: (Start)

On the infinite square grid, consider the outside corner of an infinite square.

We start at round 0 with all cells in the OFF state.

The rule: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.

At round 1, we turn ON three cells around the corner of the infinite square, forming a concave-convex hexagon with three exposed vertices.

At round 2, we turn ON nine cells around the hexagon.

At round 3, we turn ON nine other cells. Three cells around of every corner of the hexagon.

And so on.

Shows a fractal-like behavior similar to the toothpick sequence A153006.

For the first differences see the entry A162349.

For more information see A160410, which is the main entry for this sequence.

(End)

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1000

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 31.

Omar E. Pol, Illustration of initial terms [From Omar E. Pol, Nov 10 2009]

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

Index entries for sequences related to cellular automata - Omar E. Pol, Nov 10 2009

Formula

From Omar E. Pol, Nov 10 2009: (Start)

a(n) = A160410(n)*3/4.

a(0) = 0, a(n) = A130665(n-1)*3, for n>0.

(End)

Example

If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
...77..77..77..77
...766667..766667
....6556....6556.
....654444444456.
...76643344334667
...77.43222234.77
......44211244...
00000000001244...
00000000002234.77
00000000004334667
0000000000444456.
0000000000..6556.
0000000000.766667
0000000000.77..77
0000000000.......
0000000000.......
0000000000.......

Mathematica

a[n_] := 3*Sum[3^DigitCount[k, 2, 1], {k, 0, n - 1}]; Array[a, 48, 0] (* Michael De Vlieger, Nov 01 2022 *)

Crossrefs

Cf. A139250, A139251, A153006, A152980, A160410, A160414.

Cf. A130665, A162349. - Omar E. Pol, Nov 10 2009

Sequence in context: A044436 A210282 A160167 * A091846 A061262 A051656

Adjacent sequences: A160409 A160410 A160411 * A160413 A160414 A160415

Keyword

nonn

Author

Omar E. Pol, May 20 2009, Jun 01 2009

Extensions

More terms from Omar E. Pol, Nov 10 2009

Edited by Omar E. Pol, Nov 11 2009

More terms from Nathaniel Johnston, Nov 06 2010

More terms from Colin Barker, Apr 19 2015

Status

approved