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A269912 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 3", based on the 5-celled von Neumann neighborhood. 1
1, 6, 7, 52, 53, 170, 171, 392, 393, 750, 751, 1276, 1277, 2002, 2003, 2960, 2961, 4182, 4183, 5700, 5701, 7546, 7547, 9752, 9753, 12350, 12351, 15372, 15373, 18850, 18851, 22816, 22817, 27302, 27303, 32340, 32341, 37962, 37963, 44200, 44201, 51086, 51087 (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
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
FORMULA
Conjectures from Colin Barker, Mar 08 2016: (Start)
a(n) = (3*(1+(-1)^n)+(2-12*(-1)^n)*n-6*(-2+(-1)^n)*n^2+4*n^3)/6.
a(n) = (2*n^3+3*n^2-5*n+3)/3 for n even.
a(n) = (2*n^3+9*n^2+7*n)/3 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+5*x-2*x^2+30*x^3+x^4-3*x^5) / ((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=3; 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. A269910.
Sequence in context: A042843 A182622 A028423 * A042117 A335956 A179885
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 07 2016
STATUS
approved

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Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)