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A273417
Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.
4
1, 4, 21, 41, 72, 113, 160, 217, 280, 353, 432, 521, 616, 721, 832, 953, 1080, 1217, 1360, 1513, 1672, 1841, 2016, 2201, 2392, 2593, 2800, 3017, 3240, 3473, 3712, 3961, 4216, 4481, 4752, 5033, 5320, 5617, 5920, 6233, 6552, 6881, 7216, 7561, 7912, 8273, 8640
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, May 22 2016: (Start)
a(n) = (-15-(-1)^n+8*n+8*n^2)/2 for n>2.
a(n) = 4*(n^2+n-2) for n>2 and even.
a(n) = 4*n^2+4*n-7 for n>2 and odd.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>6.
G.f.: (1+2*x+13*x^2+x^3-3*x^4+7*x^5-5*x^6) / ((1-x)^3*(1+x)).
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=705; 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: A085703 A063122 A110737 * A273443 A304467 A316284
KEYWORD
nonn,easy
AUTHOR
Robert Price, May 22 2016
STATUS
approved