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A283358
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 621", based on the 5-celled von Neumann neighborhood.
4
1, 1, 1, 13, 17, 29, 49, 125, 241, 509, 1009, 2045, 4081, 8189, 16369, 32765, 65521, 131069, 262129, 524285, 1048561, 2097149, 4194289, 8388605, 16777201, 33554429, 67108849, 134217725, 268435441, 536870909, 1073741809, 2147483645, 4294967281, 8589934589
OFFSET
0,4
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 07 2017: (Start)
G.f.: (1 - x - 2*x^2 + 12*x^3 - 8*x^4 - 16*x^5 + 32*x^7) / ((1 - x)*(1 + x)*(1 - 2*x)).
a(n) = 2^n - 15 for n>4 and even.
a(n) = 2^n - 3 for n>4 and odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>5.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 621; 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, Mar 05 2017
STATUS
approved