A270218 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.

1, 4, 28, 140, 620, 2604, 10668, 43180, 173740, 697004, 2792108, 11176620, 44722860, 178924204, 715762348, 2863180460

Offset: 0

Author

Robert Price, Mar 13 2016

Keywords

• nonn
• more

Comments

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

It appears Rules 385, 425, 465 and 553 also generate this sequence. - Lars Blomberg, Apr 30 2016 (It would be nice to have a proof! - N. J. A. Sloane, May 09 2016)

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. A270217.

Programs

Mathematica

CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=129; 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

Conjectures from Colin Barker, Mar 13 2016: (Start)

a(n) = 4*(1-3*2^n+2^(1+2*n))/3.

a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.

G.f.: (1-3*x+14*x^2-8*x^3) / ((1-x)*(1-2*x)*(1-4*x)).

(End)

a(n) = 4*A006095(n+1) (conjectured). - Michal Stajszczak, May 20 2020

Extensions

a(8)-a(15) from Lars Blomberg, Apr 30 2016