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A270985 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 238", based on the 5-celled von Neumann neighborhood. 0
1, 5, 20, 76, 300, 1212, 4900, 19740, 79284, 317836, 1272804, 5094204, 20382868, 81543660, 326199108, 1304845468 (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, Jun 07 2016: (Start)
a(n) = (320-27*2^(3+n)+175*4^n+384*n)/144 for n>2.
a(n) = 8*a(n-1)-21*a(n-2)+22*a(n-3)-8*a(n-4) for n>6.
G.f.: (1+x)*(1-4*x+5*x^2-6*x^3+16*x^4-8*x^5) / ((1-x)^2*(1-2*x)*(1-4*x)).
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=238; 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 *)
Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)
CROSSREFS
Cf. A270984.
Sequence in context: A300918 A269708 A295347 * A289786 A129869 A271887
KEYWORD
nonn,more
AUTHOR
Robert Price, Mar 27 2016
EXTENSIONS
a(8)-a(15) from Lars Blomberg, Jun 07 2016
STATUS
approved

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Last modified May 27 17:52 EDT 2024. Contains 372880 sequences. (Running on oeis4.)