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A281216
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 342", based on the 5-celled von Neumann neighborhood.
4
1, 3, 4, 14, 17, 59, 68, 238, 273, 955, 1092, 3822, 4369, 15291, 17476, 61166, 69905, 244667, 279620, 978670, 1118481, 3914683, 4473924, 15658734, 17895697, 62634939, 71582788, 250539758, 286331153, 1002159035, 1145324612, 4008636142, 4581298449, 16034544571
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, Jan 18 2017: (Start)
a(n) = 4*a(n-2) + a(n-4) - 4*a(n-6) for n>5.
G.f.: (1+3*x+2*x^3) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(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 = 342; 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, Jan 17 2017
STATUS
approved