A271804 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 401", based on the 5-celled von Neumann neighborhood.

1, 4, 32, 164, 744, 3196, 13376, 55220, 226200, 922156, 3747248, 15191396, 61473672, 248391004, 1002403616, 4040994836

Offset

0,2

Comments

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

Conjecture: Rule 409 also generates this sequence. - Lars Blomberg, Jun 22 2016

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

Conjectures from Colin Barker, Jun 22 2016: (Start)

a(n) = (1-2^(2+n)+4^(1+n)+(2^(-1-n)*(-(3-sqrt(17))^n*(-7+sqrt(17))-(3+sqrt(17))^n*(7+sqrt(17))))/sqrt(17)) for n>0.

G.f.: (1-6*x+25*x^2-60*x^3+12*x^4+16*x^5) / ((1-x)*(1-2*x)*(1-4*x)*(1-3*x-2*x^2)).

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=401; 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. A271803.

Sequence in context: A272424 A269811 A271126 * A270207 A270910 A269873

Adjacent sequences: A271801 A271802 A271803 * A271805 A271806 A271807

Keyword

nonn,more

Author

Robert Price, Apr 14 2016

Extensions

a(8)-a(15) from Lars Blomberg, Jun 22 2016

Status

approved