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A270934
Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 221", based on the 5-celled von Neumann neighborhood.
4
1, 8, 4, 44, 9, 116, 9, 220, 9, 356, 9, 524, 9, 724, 9, 956, 9, 1220, 9, 1516, 9, 1844, 9, 2204, 9, 2596, 9, 3020, 9, 3476, 9, 3964, 9, 4484, 9, 5036, 9, 5620, 9, 6236, 9, 6884, 9, 7564, 9, 8276, 9, 9020, 9, 9796, 9, 10604, 9, 11444, 9, 12316, 9, 13220, 9
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.
FORMULA
Conjectures from Colin Barker, Mar 26 2016: (Start)
a(n) = (5+13*(-1)^n-4*(-1+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2 for n>2.
a(n) = 9 for n>2 and even.
a(n) = 4*n^2+4*n-4 for n>2 and odd.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>8.
G.f.: (1+8*x+x^2+20*x^3+8*x^5-7*x^6-4*x^7+5*x^8) / ((1-x)^3*(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=221; 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}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
CROSSREFS
Sequence in context: A270626 A270677 A270901 * A271004 A271051 A046106
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 26 2016
STATUS
approved