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A270681 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 190", based on the 5-celled von Neumann neighborhood. 4
1, 5, 12, 24, 32, 52, 60, 88, 96, 132, 140, 184, 192, 244, 252, 312, 320, 388, 396, 472, 480, 564, 572, 664, 672, 772, 780, 888, 896, 1012, 1020, 1144, 1152, 1284, 1292, 1432, 1440, 1588, 1596, 1752, 1760, 1924, 1932, 2104, 2112, 2292, 2300, 2488, 2496, 2692 (list; graph; refs; listen; history; text; internal format)
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

Robert Price, Diagrams of the first 20 stages.

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 21 2016: (Start)

a(n) = 3/2*(-1+(-1)^n)-(-5+(-1)^n)*n+n^2 for n>1.

a(n) = n^2+4*n for n>1 and even.

a(n) = n^2+6*n-3 for n>1 and odd.

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

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

(End)

MATHEMATICA

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

code=190; 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}];

Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

CROSSREFS

Sequence in context: A344846 A066869 A023172 * A212540 A344510 A100479

Adjacent sequences:  A270678 A270679 A270680 * A270682 A270683 A270684

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 21 2016

STATUS

approved

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Last modified July 30 23:12 EDT 2021. Contains 346365 sequences. (Running on oeis4.)