A160118 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

0, 1, 9, 13, 41, 45, 73, 85, 169, 173, 201, 213, 297, 309, 393, 429, 681, 685, 713, 725, 809, 821, 905, 941, 1193, 1205, 1289, 1325, 1577, 1613, 1865, 1973, 2729, 2733, 2761, 2773, 2857, 2869, 2953, 2989, 3241, 3253, 3337, 3373, 3625, 3661, 3913, 4021, 4777, 4789

Offset

0,3

Comments

On the infinite square grid, we start at stage 0 with all square cells in the OFF state.

Define a "peninsula cell" to a cell that is connected to the structure by exactly one of its vertices.

At stage 1 we turn ON a single cell in the central position.

For n>1, if n is even, at stage n we turn ON all the OFF neighboring cells from cells that were turned in ON at stage n-1.

For n>1, if n is odd, at stage n we turn ON all the peninsular OFF cells.

For the corresponding corner sequence, see A160796.

An animation will show the fractal-like behavior (cf. A139250).

For the first differences see A160415. - Omar E. Pol, Mar 21 2011

First differs from A188343 at a(13). - Omar E. Pol, Mar 28 2011

Links

Table of n, a(n) for n=0..49.

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.],

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

Index entries for sequences related to cellular automata.

Formula

From Nathaniel Johnston, Mar 24 2011: (Start)

a(2n-1) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n-1} A147610(k), n >= 2.

a(2n) = 9 + 4*Sum_{k=2..n} A147610(k) + 28*Sum_{k=2..n} A147610(k), n >= 1.

(End)

Example

If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...............9
.888.888.888.888.
.878.878.878.878.
.8866688.8866688.
...656.....656...
.8866444.4446688.
.878.434.434.878.
.888.4422244.888.
.......212.......
.888.4422244.888.
.878.434.434.878.
.8866444.4446688.
...656.....656...
.8866688.8866688.
.878.878.878.878.
.888.888.888.888.
9...............9
In the first generation, only the central "1" is ON, a(1)=1. In the next generation, we turn ON eight "2" around the central cell, leading to a(2)=a(1)+8=9. In the third generation, four "3" are turned ON at the vertices of the square, a(3)=a(2)+4=13. And so on...

Mathematica

With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; a[n_] := If[OddQ[n], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2)*Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]; a[0] = 0; a[1] = 1; Array[a, 50, 0]] (* Amiram Eldar, Aug 01 2023 *)

Crossrefs

Cf. A139250, A139251, A147562, A147610, A160117, A160119, A160379, A160415, A160796, A188343.

Sequence in context: A146225 A219257 A200537 * A188343 A160117 A068984

Adjacent sequences: A160115 A160116 A160117 * A160119 A160120 A160121

Keyword

nonn

Author

Omar E. Pol, May 05 2009

Extensions

Entry revised by Omar E. Pol and N. J. A. Sloane, Feb 16 2010, Feb 21 2010

a(8) - a(38) from Nathaniel Johnston, Nov 06 2010

a(13) corrected at the suggestion of Sean A. Irvine. Then I corrected 19 terms between a(14) and a(38). Finally I added a(39)-a(42). - Omar E. Pol, Mar 21 2011

Rule, for n even, edited by Omar E. Pol, Mar 22 2011

Incorrect comment (in "formula" section) removed by Omar E. Pol, Mar 23 2011, with agreement of author.

More terms from Amiram Eldar, Aug 01 2023

Status

approved