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A270012 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 7", based on the 5-celled von Neumann neighborhood. 1
1, 10, 10, 59, 59, 180, 180, 405, 405, 766, 766, 1295, 1295, 2024, 2024, 2985, 2985, 4210, 4210, 5731, 5731, 7580, 7580, 9789, 9789, 12390, 12390, 15415, 15415, 18896, 18896, 22865, 22865, 27354, 27354, 32395, 32395, 38020, 38020, 44261, 44261, 51150, 51150 (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) = 1/4*(7-3*(-1)^n)+(11/6-2*(-1)^n)*n-(-2+(-1)^n)*n^2+(2*n^3)/3.

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

a(n) = (4*n^3+18*n^2+23*n+15)/6 for n 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) for n>6.

G.f.: (1+9*x-3*x^2+22*x^3+3*x^4+x^5-x^6) / ((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=7; 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. A270010.

Sequence in context: A238017 A111220 A106789 * A219911 A056483 A056473

Adjacent sequences:  A270009 A270010 A270011 * A270013 A270014 A270015

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 08 2016

STATUS

approved

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Last modified April 6 16:14 EDT 2020. Contains 333276 sequences. (Running on oeis4.)