A272743 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 526", based on the 5-celled von Neumann neighborhood.

1, 5, 17, 69, 277, 1109, 4437, 17749, 70997, 283989, 1135957, 4543829, 18175317, 72701269, 290805077, 1163220309

Offset

0,2

Comments

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

References

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

Links

Table of n, a(n) for n=0..15.

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

Formula

Conjecture: a(n) = (13*4^(n-1) - 1)/3, n>1. - Lars Blomberg, Jul 08 2016

Conjectures from Colin Barker, Jul 08 2016: (Start)

a(n) = 5*a(n-1)-4*a(n-2) for n>4.

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

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=526; 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 *)

Crossrefs

Cf. A272742.

Sequence in context: A149698 A149699 A151276 * A149700 A149701 A149702

Adjacent sequences: A272740 A272741 A272742 * A272744 A272745 A272746

Keyword

nonn,more

Author

Robert Price, May 05 2016

Extensions

a(8)-a(15) from Lars Blomberg, Jul 08 2016

Status

approved