A284244 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 841", based on the 5-celled von Neumann neighborhood.

1, 0, 110, 1110, 11110, 101110, 101110, 10111110, 11111110, 1110101110, 11010101110, 101010111110, 101011111110, 10101110101110, 10101010101110, 1010101010111110, 1010101011111110, 101010101110101110, 101010101010101110, 10101010101010111110

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 06 2024: (Start)

a(n) = 100*a(n-2) + a(n-4) - 100*a(n-6) for n > 16.

G.f.: (1000000000000*x^16 - 10000000000*x^14 - 1000000000000*x^12 - 10000000000*x^11 + 9900000000*x^10 + 99000000*x^9 + 1000000*x^8 - 1000*x^7 - 1009900*x^6 - 9890*x^5 + 109*x^4 + 1110*x^3 + 10*x^2 + 1)/(100*x^6 - x^4 - 100*x^2 + 1). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 841; 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[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Crossrefs

Cf. A284243, A284245, A284246.

Sequence in context: A283219 A283137 A213087 * A232781 A248468 A231967

Adjacent sequences: A284241 A284242 A284243 * A284245 A284246 A284247

Keyword

nonn,easy

Author

Robert Price, Mar 23 2017

Status

approved