A281748 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 395", based on the 5-celled von Neumann neighborhood.

1, 1, 111, 100, 11101, 10111, 1110100, 1010101, 111010111, 101110100, 11101010101, 10101110111, 1110101010100, 1011101110101, 111010101010111, 101011101110100, 11101010101010101, 10111010101110111, 1110101010101010100, 1010111010101110101

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

Conjectures from Chai Wah Wu, May 05 2024: (Start)

a(n) = - a(n-1) + a(n-3) + a(n-4) + 100000000*a(n-8) + 100000000*a(n-9) - 100000000*a(n-11) - 100000000*a(n-12) for n > 19.

G.f.: (-1000000000*x^19 - 1000000000*x^18 - 1000000000*x^17 + 1000000000*x^15 + 90000000*x^12 + 90100000*x^11 + 11000000*x^9 + 11999000*x^8 + 2109000*x^7 + 1120000*x^6 + 21100*x^5 + 11199*x^4 + 210*x^3 + 112*x^2 + 2*x + 1)/(100000000*x^12 + 100000000*x^11 - 100000000*x^9 - 100000000*x^8 - x^4 - x^3 + x + 1). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 395; 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. A281749, A281750, A281751.

Sequence in context: A282359 A282447 A282606 * A282363 A281894 A282103

Adjacent sequences: A281745 A281746 A281747 * A281749 A281750 A281751

Keyword

nonn,easy

Author

Robert Price, Jan 29 2017

Status

approved