A271062 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 261", based on the 5-celled von Neumann neighborhood.

1, 9, 10, 58, 59, 179, 180, 404, 405, 765, 766, 1294, 1295, 2023, 2024, 2984, 2985, 4209, 4210, 5730, 5731, 7579, 7580, 9788, 9789, 12389, 12390, 15414, 15415, 18895, 18896, 22864, 22865, 27353, 27354, 32394, 32395, 38019, 38020, 44260, 44261, 51149, 51150

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 Chai Wah Wu, Dec 29 2016: (Start)

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n > 6.

G.f.: (x^4 + 24*x^3 - 2*x^2 + 8*x + 1)/((x - 1)^4*(x + 1)^3). (End)

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 *)
Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)

Crossrefs

Cf. A271060.

Sequence in context: A259914 A368047 A037954 * A041174 A048070 A341527

Adjacent sequences: A271059 A271060 A271061 * A271063 A271064 A271065

Keyword

nonn,easy

Author

Robert Price, Mar 29 2016

Status

approved