A282105 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 427", based on the 5-celled von Neumann neighborhood.

1, 1, 7, 4, 29, 23, 124, 69, 471, 380, 1989, 1111, 7548, 6085, 31831, 17788, 120773, 97367, 509308, 284613, 1932375, 1557884, 8148933, 4553815, 30918012, 24926149, 130382935, 72861052, 494688197, 398818391, 2086126972, 1165776837, 7915011159, 6381094268

Offset

0,3

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

Empirical g.f.: (1 + x + 7*x^2 + 3*x^3 + 12*x^4 + 8*x^6 - 8*x^7) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 + x + x^2)*(1 + 4*x^2)). - Colin Barker, Feb 06 2017

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 427; 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. A282103, A282104, A282106.

Sequence in context: A281750 A282365 A281896 * A282259 A282453 A282654

Adjacent sequences: A282102 A282103 A282104 * A282106 A282107 A282108

Keyword

nonn,easy

Author

Robert Price, Feb 06 2017

Status

approved