A282956 Decimal 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, 2, 4, 9, 18, 36, 75, 145, 300, 579, 1213, 2322, 4800, 9279, 19421, 37162, 77008, 148007, 310745, 594598, 1232216, 2368039, 4971993, 9513126, 19715928, 37888551, 79551961, 152209574, 315455320, 606216743, 1272831449, 2435352742, 5047285592, 9699467815

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) = 17*a(n-4) - 16*a(n-8) for n > 24.

G.f.: (384*x^24 - 448*x^23 + 64*x^22 + 384*x^21 - 120*x^20 - 4*x^19 - 4*x^18 - 472*x^17 + 208*x^16 + 8*x^15 + 12*x^13 - 12*x^12 + x^11 + 2*x^10 - x^9 + 10*x^8 - 8*x^7 + 7*x^6 + 2*x^5 + x^4 + 9*x^3 + 4*x^2 + 2*x + 1)/(16*x^8 - 17*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]], 2], {i, 1, stages - 1}]

Crossrefs

Cf. A282954, A282955, A282957.

Sequence in context: A293327 A282909 A282825 * A282986 A056185 A363262

Adjacent sequences: A282953 A282954 A282955 * A282957 A282958 A282959

Keyword

nonn,easy

Author

Robert Price, Feb 25 2017

Status

approved