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A284181
Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 813", based on the 5-celled von Neumann neighborhood.
4
1, 2, 5, 10, 23, 47, 87, 171, 381, 766, 1407, 2751, 6111, 12287, 22527, 44031, 97791, 196607, 360447, 704511, 1564671, 3145727, 5767167, 11272191, 25034751, 50331647, 92274687, 180355071, 400556031, 805306367, 1476395007, 2885681151, 6408896511, 12884901887
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 22 2017: (Start)
G.f.: (1 + x + 3*x^2 + 5*x^3 - 3*x^4 + 8*x^5 - 8*x^6 + 4*x^7 + 2*x^8 + x^9 + x^10 + 16*x^13 - 16*x^14) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)).
a(n) = a(n-1) + 16*a(n-4) - 16*a(n-5) for n>10.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 813; 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}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[1, i]], 2], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 21 2017
STATUS
approved