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A273311 Partial sums of the 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. 1
1, 5, 22, 62, 135, 247, 408, 624, 905, 1257, 1690, 2210, 2827, 3547, 4380, 5332, 6413, 7629, 8990, 10502, 12175, 14015, 16032, 18232, 20625, 23217, 26018, 29034, 32275, 35747, 39460, 43420, 47637, 52117, 56870, 61902, 67223, 72839, 78760, 84992, 91545, 98425 (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, May 19 2016: (Start)

a(n) = (57+3*(-1)^n-58*n+48*n^2+16*n^3)/12 for n>0.

a(n) = (8*n^3+24*n^2-29*n+30)/6 for n>0 and even.

a(n) = (8*n^3+24*n^2-29*n+27)/6 for n>0 and odd.

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>5.

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

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. A273309.

Sequence in context: A050533 A212094 A064836 * A288534 A286711 A222632

Adjacent sequences:  A273308 A273309 A273310 * A273312 A273313 A273314

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 19 2016

STATUS

approved

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Last modified January 26 17:27 EST 2020. Contains 331280 sequences. (Running on oeis4.)