A282954 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 533", based on the 5-celled von Neumann neighborhood.

1, 10, 100, 1001, 10010, 100100, 1001011, 10010001, 100101100, 1001000011, 10010111101, 100100010010, 1001011000000, 10010000111111, 100101111011101, 1001000100101010, 10010110011010000, 100100001000100111, 1001011110111011001, 10010001001010100110

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.

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 Chai Wah Wu, May 05 2024: (Start)

a(n) = 10001*a(n-4) - 10000*a(n-8) for n > 24.

G.f.: (990000000*x^24 - 991000000*x^23 + 1000000*x^22 + 110000000*x^21 - 9999000*x^20 - 900*x^19 - 100*x^18 - 111011000*x^17 + 11010000*x^16 + 1000*x^15 + 1100*x^13 - 1100*x^12 + 9*x^11 + 90*x^10 - 89*x^9 + 1090*x^8 - 1000*x^7 + 911*x^6 + 90*x^5 + 9*x^4 + 1001*x^3 + 100*x^2 + 10*x + 1)/(10000*x^8 - 10001*x^4 + 1). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 533; 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. A282955, A282956, A282957.

Sequence in context: A033146 A282907 A281730 * A282984 A266071 A118256

Adjacent sequences: A282951 A282952 A282953 * A282955 A282956 A282957

Keyword

nonn,easy

Author

Robert Price, Feb 25 2017

Status

approved