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A271091 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 275", based on the 5-celled von Neumann neighborhood. 4
1, 5, 5, 40, 5, 112, 5, 216, 5, 352, 5, 520, 5, 720, 5, 952, 5, 1216, 5, 1512, 5, 1840, 5, 2200, 5, 2592, 5, 3016, 5, 3472, 5, 3960, 5, 4480, 5, 5032, 5, 5616, 5, 6232, 5, 6880, 5, 7560, 5, 8272, 5, 9016, 5, 9792, 5, 10600, 5, 11440, 5, 12312, 5, 13216, 5 (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 30 2016: (Start)

a(n) = (-3+13*(-1)^n-4*(-1+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2 for n>1.

a(n) = 5 for n>1 and even.

a(n) = 4*n^2+4*n-8 for n>1 and odd.

a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>7.

G.f.: (1+5*x+2*x^2+25*x^3-7*x^4+7*x^5+4*x^6-5*x^7) / ((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=275; 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: A139386 A074947 A099757 * A271289 A271083 A271277

Adjacent sequences:  A271088 A271089 A271090 * A271092 A271093 A271094

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 30 2016

STATUS

approved

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Last modified September 30 10:16 EDT 2020. Contains 337438 sequences. (Running on oeis4.)