A270089 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 73", based on the 5-celled von Neumann neighborhood.

1, 5, 10, 50, 50, 171, 171, 396, 396, 757, 757, 1286, 1286, 2015, 2015, 2976, 2976, 4201, 4201, 5722, 5722, 7571, 7571, 9780, 9780, 12381, 12381, 15406, 15406, 18887, 18887, 22856, 22856, 27345, 27345, 32386, 32386, 38011, 38011, 44252, 44252, 51141, 51141

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, Mar 11 2016: (Start)

a(n) = (-87-9*(-1)^n+(22-24*(-1)^n)*n-12*(-2+(-1)^n)*n^2+8*n^3)/12 for n>2.

a(n) = (4*n^3+6*n^2-n-48)/6 for n>2 and even.

a(n) = (4*n^3+18*n^2+23*n-39)/6 for n>2 and odd.

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>9.

G.f.: (1+4*x+2*x^2+28*x^3-12*x^4+13*x^5+14*x^6-22*x^7-5*x^8+9*x^9) / ((1-x)^4*(1+x)^3).

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=73; 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. A270087.

Sequence in context: A034190 A216390 A305476 * A336582 A271275 A270100

Adjacent sequences: A270086 A270087 A270088 * A270090 A270091 A270092

Keyword

nonn,easy

Author

Robert Price, Mar 10 2016

Status

approved