A273314 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

1, 6, 23, 64, 137, 250, 411, 628, 909, 1262, 1695, 2216, 2833, 3554, 4387, 5340, 6421, 7638, 8999, 10512, 12185, 14026, 16043, 18244, 20637, 23230, 26031, 29048, 32289, 35762, 39475, 43436, 47653, 52134, 56887, 61920, 67241, 72858, 78779, 85012, 91565, 98446

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

Robert Price, Table of n, a(n) for n = 0..128

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, May 19 2016: (Start)

a(n) = (4*n^3+12*n^2-13*n+15)/3 for n>0.

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.

G.f.: (1+2*x+5*x^2+4*x^3-4*x^4) / (1-x)^4.

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=643; 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 *)
Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)

Crossrefs

Cf. A166147.

Sequence in context: A208598 A327794 A119712 * A281424 A005745 A332081

Adjacent sequences: A273311 A273312 A273313 * A273315 A273316 A273317

Keyword

nonn,easy

Author

Robert Price, May 19 2016

Status

approved