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A280610
Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 294", based on the 5-celled von Neumann neighborhood.
4
1, 3, 1, 7, 17, 39, 65, 135, 257, 519, 1025, 2055, 4097, 8199, 16385, 32775, 65537, 131079, 262145, 524295, 1048577, 2097159, 4194305, 8388615, 16777217, 33554439, 67108865, 134217735, 268435457, 536870919, 1073741825, 2147483655, 4294967297, 8589934599
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
Empirical g.f.: (1 + x - 6*x^2 + 4*x^3 + 8*x^4 - 16*x^6)/((1 - x)*(1 + x)*(1 - 2*x)). - Ilya Gutkovskiy, Jan 06 2017
Conjectures from Colin Barker, Jan 07 2017: (Start)
a(n) = 4 - 3*(-1)^n + 2^n for n>3.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>4.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 294; 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[i, 2 * i - 1]], 2], {i , 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jan 06 2017
STATUS
approved