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A283407
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 637", based on the 5-celled von Neumann neighborhood.
4
1, 1, 1, 13, 17, 29, 49, 125, 113, 893, 1137, 2045, 4081, 8189, 16369, 32765, 65521, 131069, 262129, 524285, 1048561, 2097149, 4194289, 8388605, 16777201, 33554429, 67108849, 134217725, 268435441, 536870909, 536870897, 4294967293, 8589934577, 17179869181
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 Chai Wah Wu, May 06 2024: (Start)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n > 33.
G.f.: (-3221225472*x^33 + 536870912*x^32 + 3221225472*x^31 - 536870912*x^30 + 256*x^13 + 640*x^12 - 896*x^11 - 512*x^10 + 640*x^9 - 128*x^8 + 32*x^7 - 16*x^5 - 8*x^4 + 12*x^3 - 2*x^2 - x + 1)/((x - 1)*(x + 1)*(2*x - 1)). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 637; 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 07 2017
STATUS
approved