A284300 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 865", based on the 5-celled von Neumann neighborhood.

1, 0, 110, 1010, 1010, 101110, 101010, 11101110, 110101010, 1011101110, 1010101010, 101011101110, 101010101010, 10101011101110, 10101010101010, 1010101111101110, 1010101110101010, 101011101011101110, 101010101010101010, 10111110101011101110

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) = 100*a(n-2) + a(n-8) - 100*a(n-10) for n > 36.

G.f.: (10000000000000000000000*x^36 + 1000000000000000000000000*x^35 + 1009900000000000000000000*x^34 - 10000000000000000000000*x^33 + 990000000000000000000000*x^32 + 10000000000000000000000*x^31 - 10000000000000000000000*x^30 - 99000000000000000000*x^29 - 10000010000000000000000*x^28 - 1000000010000000000000000*x^27 - 1009900000000000000000000*x^26 + 10000000000000000000000*x^25 - 990000000000000000000000*x^24 - 10000000000000000000000*x^23 + 10000000000000000000000*x^22 + 98990000000000000000*x^21 + 10000000000000000*x^20 + 10000000000000000*x^19 + 990000000000*x^17 - 99010000*x^13 + 10000*x^12 - 99010900*x^11 - 10000000000*x^10 - 99009890*x^9 + 100000009*x^8 + 990110*x^7 + 10*x^6 + 110*x^5 - 9990*x^4 + 1010*x^3 + 10*x^2 + 1)/(100*x^10 - x^8 - 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 = 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[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Crossrefs

Cf. A284299, A284301, A284302.

Sequence in context: A283401 A283348 A284236 * A284348 A234760 A234753

Adjacent sequences: A284297 A284298 A284299 * A284301 A284302 A284303

Keyword

nonn,easy

Author

Robert Price, Mar 24 2017

Status

approved