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A283646
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 694", based on the 5-celled von Neumann neighborhood.
4
1, 3, 5, 13, 21, 53, 93, 205, 349, 845, 1501, 3277, 5597, 13517, 24029, 52685, 89565, 216525, 384477, 839117, 1433053, 3460557, 6151645, 13487565, 22928861, 55430605, 98426333, 214814157, 366861789, 885902797, 1574821341, 3452816845, 5869788637, 14190235085
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 14 2017: (Start)
G.f.: (1 + 3*x + 4*x^2 + 10*x^3 + 16*x^4 + 40*x^5 + 72*x^6 + 152*x^7 - 128*x^9 + 128*x^10 - 128*x^11 + 256*x^15) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)*(1 + 16*x^4)).
a(n) = a(n-2) + 256*a(n-8) - 256*a(n-10) for n>11.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 694; 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 12 2017
STATUS
approved