A284299 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 865", based on the 5-celled von Neumann neighborhood.

1, 0, 11, 101, 1010, 11101, 101010, 1110111, 10101011, 111011101, 1010101010, 11101110101, 101010101010, 1110111010101, 10101010101010, 111011111010101, 1010101110101010, 11101110101110101, 101010101010101010, 1110111010101111101, 10101010101010111010

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 > 36.

G.f.: (100000000000000*x^36 + 100000000000*x^35 - 98990000000000*x^34 - 100000000000*x^33 - 9900000000*x^32 + 1000000000*x^31 - 100000000*x^30 + 99000000000*x^29 - 1000001000000*x^28 - 100000001000*x^27 + 989900*x^26 + 1000*x^25 + 99*x^24 - 10*x^23 + x^22 - 100990*x^21 + 10000*x^20 + 1000*x^19 - 9900000*x^17 - 990100000*x^13 + 100000000*x^12 + 890099000*x^11 - x^10 + 109900990*x^9 - 89999999*x^8 + 1099010*x^7 + 100000*x^6 + 11000*x^5 + 999*x^4 + 101*x^3 + 10*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 = 865; 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. A284300, A284301, A284302.

Sequence in context: A089183 A164046 A284235 * A284347 A113999 A283062

Adjacent sequences: A284296 A284297 A284298 * A284300 A284301 A284302

Keyword

nonn,easy,changed

Author

Robert Price, Mar 24 2017

Status

approved