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A273144
Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 597", based on the 5-celled von Neumann neighborhood.
4
1, 8, 20, 41, 57, 104, 116, 193, 193, 312, 292, 457, 409, 632, 548, 833, 705, 1064, 884, 1321, 1081, 1608, 1300, 1921, 1537, 2264, 1796, 2633, 2073, 3032, 2372, 3457, 2689, 3912, 3028, 4393, 3385, 4904, 3764, 5441, 4161, 6008, 4580, 6601, 5017, 7224, 5476
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 16 2016: (Start)
a(n) = 2*a(n-2)-2*a(n-6)+a(n-8) for n>7.
G.f.: (1+8*x+18*x^2+25*x^3+17*x^4+22*x^5+4*x^6+x^7) / ((1-x)^3*(1+x)^3*(1+x^2)).
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=597; 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: A006416 A375717 A273069 * A272842 A272924 A272943
KEYWORD
nonn,easy
AUTHOR
Robert Price, May 16 2016
STATUS
approved