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A269906 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 1", based on the 5-celled von Neumann neighborhood. 4
1, 4, 1, 44, 1, 116, 1, 220, 1, 356, 1, 524, 1, 724, 1, 956, 1, 1220, 1, 1516, 1, 1844, 1, 2204, 1, 2596, 1, 3020, 1, 3476, 1, 3964, 1, 4484, 1, 5036, 1, 5620, 1, 6236, 1, 6884, 1, 7564, 1, 8276, 1, 9020, 1, 9796, 1, 10604, 1, 11444, 1, 12316, 1, 13220, 1 (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 08 2016: (Start)

a(n) = (-3+5*(-1)^n-4*(-1+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2.

a(n) = 1 for n even.

a(n) = 4*n^2+4*n-4 for n odd.

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

G.f.: (1+4*x-2*x^2+32*x^3+x^4-4*x^5) / ((1-x)^3*(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=1; 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: A298495 A193962 A302441 * A092667 A060627 A113101

Adjacent sequences:  A269903 A269904 A269905 * A269907 A269908 A269909

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 07 2016

STATUS

approved

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Last modified December 15 22:02 EST 2019. Contains 330012 sequences. (Running on oeis4.)