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A270008 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 5", based on the 5-celled von Neumann neighborhood. 1
1, 9, 9, 58, 58, 179, 179, 404, 404, 765, 765, 1294, 1294, 2023, 2023, 2984, 2984, 4209, 4209, 5730, 5730, 7579, 7579, 9788, 9788, 12389, 12389, 15414, 15414, 18895, 18895, 22864, 22864, 27353, 27353, 32394, 32394, 38019, 38019, 44260, 44260, 51149, 51149 (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

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 09 2016: (Start)

a(n) = (9-9*(-1)^n+(22-24*(-1)^n)*n-12*(-2+(-1)^n)*n^2+8*n^3)/12 for n>0.

a(n) = (4*n^3+6*n^2-n)/6 for n>0 and even.

a(n) = (4*n^3+18*n^2+23*n+9)/6 for n>0 and odd.

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).

G.f.: (1+8*x-3*x^2+25*x^3+3*x^4-2*x^5-x^6+x^7) / ((1-x)^4*(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=5; 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}];

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

Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)

CROSSREFS

Cf. A270006.

Sequence in context: A145971 A241868 A243125 * A255743 A183893 A210052

Adjacent sequences:  A270005 A270006 A270007 * A270009 A270010 A270011

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 08 2016

STATUS

approved

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Last modified December 9 03:27 EST 2019. Contains 329872 sequences. (Running on oeis4.)