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A270164 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 107", based on the 5-celled von Neumann neighborhood. 4
1, 5, 5, 41, 12, 109, 16, 197, 40, 321, 60, 469, 84, 645, 112, 849, 144, 1081, 180, 1341, 220, 1629, 264, 1945, 312, 2289, 364, 2661, 420, 3061, 480, 3489, 544, 3945, 612, 4429, 684, 4941, 760, 5481, 840, 6049, 924, 6645, 1012, 7269, 1104, 7921, 1200, 8601 (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 12 2016: (Start)

a(n) = 1/4*(3-3*(-1)^n+(10-6*(-1)^n)*n+(8-6*(-1)^n)*n^2) for n>7.

a(n) = (n^2+2*n)/2 for n>7 and even.

a(n) = (7*n^2+8*n+3)/2 for n>7 and odd.

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>13.

G.f.: (1+5*x+2*x^2+26*x^3+x^5-6*x^6-12*x^7+23*x^8+16*x^9-24*x^10-12*x^11 +8*x^12 +4*x^13) / ((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=107; 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: A269829 A270058 A270156 * A270181 A270210 A271117

Adjacent sequences:  A270161 A270162 A270163 * A270165 A270166 A270167

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 12 2016

STATUS

approved

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Last modified August 12 08:51 EDT 2020. Contains 336438 sequences. (Running on oeis4.)