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A273309 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 641", based on the 5-celled von Neumann neighborhood. 4
1, 4, 17, 40, 73, 112, 161, 216, 281, 352, 433, 520, 617, 720, 833, 952, 1081, 1216, 1361, 1512, 1673, 1840, 2017, 2200, 2393, 2592, 2801, 3016, 3241, 3472, 3713, 3960, 4217, 4480, 4753, 5032, 5321, 5616, 5921, 6232, 6553, 6880, 7217, 7560, 7913, 8272, 8641 (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, May 19 2016: (Start)

a(n) = (-15+(-1)^n+8*n+8*n^2)/2 for n>1.

a(n) = 4*n^2+4*n-7 for n>1 and even.

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

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

G.f.: (1+2*x+9*x^2+8*x^3-4*x^5) / ((1-x)^3*(1+x)).

(End)

MATHEMATICA

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

code=641; 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: A178947 A041859 A022266 * A145995 A275815 A018973

Adjacent sequences:  A273306 A273307 A273308 * A273310 A273311 A273312

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 19 2016

STATUS

approved

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Last modified November 13 12:45 EST 2019. Contains 329094 sequences. (Running on oeis4.)