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A273336 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood. 1
1, 5, 22, 70, 150, 270, 438, 662, 950, 1310, 1750, 2278, 2902, 3630, 4470, 5430, 6518, 7742, 9110, 10630, 12310, 14158, 16182, 18390, 20790, 23390, 26198, 29222, 32470, 35950, 39670, 43638, 47862, 52350, 57110, 62150, 67478, 73102, 79030, 85270, 91830, 98718 (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 20 2016: (Start)

a(n) = 2/3*(2*n^3+6*n^2+4*n-15) for n>1.

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

G.f.: (1+x+8*x^2+8*x^3-17*x^4+7*x^5) / (1-x)^4.

(End)

MATHEMATICA

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

code=657; 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. A273334.

Sequence in context: A288534 A286711 A222632 * A273768 A032168 A246211

Adjacent sequences:  A273333 A273334 A273335 * A273337 A273338 A273339

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 20 2016

STATUS

approved

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Last modified January 19 14:53 EST 2020. Contains 331049 sequences. (Running on oeis4.)