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.

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.

Links

Robert Price, Table of n, a(n) for n = 0..126

Robert Price, Diagrams of first 20 stages

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Wolfram Research, Wolfram Atlas of Simple Programs

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

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

Cf. A282827, A282828, A282829.

Sequence in context: A134554 A084514 A084524 * A283914 A283065 A284183

Adjacent sequences: A282823 A282824 A282825 * A282827 A282828 A282829

Keyword

nonn,easy

Author

Robert Price, Feb 22 2017

Status

approved