A279698 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 185", based on the 5-celled von Neumann neighborhood.

1, 0, 101, 0, 10111, 100, 1010101, 10100, 101000001, 1011100, 10101010101, 101001000, 1010001001011, 10101000010, 101010001011000, 1010101011011, 10100010101011110, 101010101011110, 1010100010101010110, 10101010101010110, 101000100010101010110

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-2) + 100000000*a(n-8) - 100000000*a(n-10) for n > 27.

G.f.: (100000000000*x^27 + 100000000000*x^26 - 9900000000*x^25 - 11000000000*x^24 - 1100100000000*x^23 - 998900000000*x^22 + 99000000000*x^21 + 909000000000*x^20 + 1009999999000*x^19 - 1010000001000*x^18 - 99999999901*x^17 + 10100000110*x^16 + 11001*x^15 + 1000009989*x^14 - 990*x^13 - 1100009090*x^12 + 99989900*x^11 + 10100*x^10 + 1001000*x^9 - 10100*x^8 + 10000*x^7 + 999990*x^6 + 100*x^5 + 10010*x^4 + 100*x^2 + 1)/(100000000*x^10 - 100000000*x^8 - 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 = 185; 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. A279699, A279700, A279701.

Sequence in context: A284175 A279498 A279545 * A279601 A279499 A279546

Adjacent sequences: A279695 A279696 A279697 * A279699 A279700 A279701

Keyword

nonn,easy

Author

Robert Price, Dec 17 2016

Status

approved