login
A282826
Binary 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 521", based on the 5-celled von Neumann neighborhood.
4
1, 0, 1, 0, 111, 111, 10101, 0, 1111101, 1111100, 101010101, 0, 11111101101, 11110101100, 1010110001101, 110101100, 111110110001101, 111110110001100, 10101010110001101, 110001100, 1111110110110001101, 1111010110110001100, 101011000110110001101
OFFSET
0,5
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, Feb 22 2017: (Start)
a(n) = a(n-2) + 100000000*a(n-8) - 100000000*a(n-10) for n>15.
G.f.: (1 + 110*x^4 + 111*x^5 + 9990*x^6 - 111*x^7 - 98899000*x^8 + 1111100*x^9 + 99899000*x^10 - 1111100*x^11 + 10091000*x^12 + 10101100*x^13 - 1100000*x^14 + 100000000*x^15 - 100000*x^17) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)*(1 + 100*x^2)*(1 + 10000*x^4)).
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 521; 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]], 10], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Feb 22 2017
STATUS
approved