A279141 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 137", based on the 5-celled von Neumann neighborhood.

1, 0, 101, 0, 10101, 0, 1010101, 0, 101011101, 10100, 10101000001, 11100, 1010111010101, 101000000, 101010000010101, 111000000, 10101110101010101, 1010000000000, 1010100000111010101, 1110101000000, 101011101010000010101, 10100000111000000

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 04 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 > 15.

G.f.: (-11100*x^15 - 1000*x^14 + 111000000*x^13 + 10000000*x^12 + 11100*x^9 + 1001000*x^8 + 1000000*x^7 + 1010000*x^6 + 10000*x^5 + 10100*x^4 + 100*x^3 + 101*x^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 = 137; 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. A279142, A279143, A279144.

Sequence in context: A126686 A279028 A282088 * A279142 A281739 A281740

Adjacent sequences: A279138 A279139 A279140 * A279142 A279143 A279144

Keyword

nonn,easy

Author

Robert Price, Dec 06 2016

Status

approved