A272566 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 501", based on the 5-celled von Neumann neighborhood.

1, 9, 14, 58, 71, 183, 208, 416, 453, 793, 842, 1346, 1407, 2107, 2180, 3108, 3193, 4381, 4478, 5958, 6067, 7871, 7992, 10152, 10285, 12833, 12978, 15946, 16103, 19523, 19692, 23596, 23777, 28197, 28390, 33358, 33563, 39111, 39328, 45488, 45717, 52521, 52762

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 03 2016: (Start)

a(n) = 1/4*(75-7*(-1)^n)-(25*n)/6-(-3+(-1)^n)*n^2+(2*n^3)/3 for n>4.

a(n) = (4*n^3+12*n^2-25*n+102)/6 for n>4 and even.

a(n) = (4*n^3+24*n^2-25*n+123)/6 for n>4 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+8*x+2*x^2+20*x^3+x^4+4*x^5-4*x^7-4*x^8+8*x^9-4*x^11) / ((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=501; 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. A272564.

Sequence in context: A271814 A272418 A272155 * A271691 A272113 A272293

Adjacent sequences: A272563 A272564 A272565 * A272567 A272568 A272569

Keyword

nonn,easy

Author

Robert Price, May 02 2016

Status

approved