A271061 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 261", based on the 5-celled von Neumann neighborhood.

1, 8, 48, 224, 960, 3968, 16128, 65024, 261120, 1046528, 4190208, 16769024, 67092480, 268402688, 1073676288, 4294836224

Offset: 0

Author

Robert Price, Mar 29 2016

Keywords

• nonn
• more

Comments

Initialized with a single black (ON) cell at stage zero.

Similar to A211012.

It is conjectured that Rules 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 373, 381, 413, 445, 477, 509, 645, 653, 661, 669, 677, 685, 693, 701, 709, 717, 725, 733, 741, 749, 757 and 765 also produces this sequence. It would be nice to have a proof.

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Index entries for sequences related to cellular automata
• Index to 2D 5-Neighbor Cellular Automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A211012, A271060.

Programs

Mathematica

CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=261; stages=128;
rule=IntegerDigits[code,2,10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
PrependTo[ca,a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

Formula

Conjecture: a(n) = 4*4^n - 4*2^n, n>0. - Lars Blomberg, Jun 09 2016

Conjectures from Colin Barker, Dec 01 2016: (Start)

a(n) = 6*a(n-1) - 8*a(n-2) for n>2.

G.f.: (1 + 2*x + 8*x^2) / ((1 - 2*x) * (1 - 4*x)). (End)

Conjectured e.g.f.: (1 - 2*exp(2*x))^2. - Stefano Spezia, Mar 12 2025

Extensions

a(8)-a(15) from Lars Blomberg, Jun 09 2016