A160117 Number of "ON" cells after n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

0, 1, 9, 13, 41, 49, 101, 113, 189, 205, 305, 325, 449, 473, 621, 649, 821, 853, 1049, 1085, 1305, 1345, 1589, 1633, 1901, 1949, 2241, 2293, 2609, 2665, 3005, 3065, 3429, 3493, 3881, 3949, 4361, 4433, 4869, 4945, 5405, 5485, 5969, 6053, 6561, 6649, 7181, 7273

Offset: 0

Author

Omar E. Pol, May 05 2009, May 15 2009

Keywords

• nonn

Comments

Define "peninsula cell" to be the "ON" cell connected to the structure by exactly one of its vertices.

Define "bridge cell" to be the "ON" cell connected to two cells of the structure by exactly consecutive two of its vertices.

On the infinite square grid, we start at stage 0 with all cells in OFF state. At stage 1, we turn ON a single cell, in the central position.

In order to construct this sequence we use the following rules:

- If n is even, we turn "ON" the cells around the cells turned "ON" at the generation n-1.

- If n is odd, we turn "ON" the possible bridge cells and the possible peninsula cells.

- Everything that is already ON remains ON.

A160411, the first differences, gives the number of cells turned "ON" at n-th stage.

Examples

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...9...9...9...9
.888.888.888.888.
.878.878.878.878.
.886668666866688.
9..656.656.656..9
.886644464446688.
.878.434.434.878.
.886644222446688.
9..656.212.656..9
.886644222446688.
.878.434.434.878.
.886644464446688.
9..656.656.656..9
.886668666866688.
.878.878.878.878.
.888.888.888.888.
9...9...9...9...9
At the first generation, only the central "1" is ON, so a(1) = 1. At the second generation, we turn ON eight cells around the central cell, leading to a(2) = a(1)+8 = 9. At the third generation, we turn ON four peninsula cells, so a(3) = a(2)+4 = 13. At the fourth generation, we turn ON the cells around the cells turned ON at the third generation, so a(4) = a(3)+28 = 41. At the 5th generation, we turn ON four peninsula cells and four bridge cells, so a(5) = a(4)+8 = 49.
		

Links

• Alois P. Heinz, 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.]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A139250, A139251, A147562, A160118, A160119, A160379.

Programs

Maple

a:= proc(n) local r;
      r:= irem(n, 2);
      `if`(n<2, n, 5+(n-r)*((7*n-3*r)/2-5))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 16 2011

Mathematica

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], (7n^2 - 10n + 10)/2, (7n^2 - 20n + 23)/2]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 16 2015, after Nathaniel Johnston *)

Formula

a(2n) = 5 + 2n(7n-5) for n>=1, a(2n+1) = 5 + 2n(7n-3) for n>=1. - Nathaniel Johnston, Nov 06 2010

G.f.: x*(x^2+1)*(4*x^3+x^2+8*x+1)/((x+1)^2*(1-x)^3). - Alois P. Heinz, Sep 16 2011

Extensions

a(10) - a(27) from Nathaniel Johnston, Nov 06 2010

a(28) - a(47) from Alois P. Heinz, Sep 16 2011