A284025 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 782", based on the 5-celled von Neumann neighborhood.

1, 11, 111, 1011, 10011, 110111, 1111111, 11101011, 111010011, 1011110111, 10110111111, 101001101011, 1010010010011, 10110001110111, 100110011111111, 1101100011101011, 11111000011010011, 111010000111110111, 1110100011010111111, 10101000100001101011

Offset

0,2

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) = 10*a(n-1) - 100*a(n-2) + 1000*a(n-3) + a(n-8) - 10*a(n-9) + 100*a(n-10) - 1000*a(n-11) for n > 32.

G.f.: (100000000000000000000*x^32 + 101000000000000000000*x^30 + 1000000000000000000*x^28 + 1010000000000000000*x^27 - 10000000000000000*x^26 - 9900000000000000*x^25 - 100010001000000000000*x^24 - 101000000000000*x^23 - 101000099010000000000*x^22 - 1010000000000*x^21 - 999999010000000000*x^20 - 1010000000000000000*x^19 + 9999999900000000*x^18 + 9899990100000000*x^17 + 999900000000*x^16 + 100990000000000*x^15 - 990100000000*x^14 - 100000000*x^13 - 100000000*x^12 - 9999010000*x^11 - 1000000*x^10 - 99999900*x^9 + 1000000*x^8 + 990001*x^7 + 101*x^6 + 101*x^5 + x^4 + x^3 + 101*x^2 + x + 1)/((x - 1)*(x + 1)*(10*x - 1)*(x^2 + 1)*(100*x^2 + 1)*(x^4 + 1)). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 782; 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. A284024, A284026, A284027.

Sequence in context: A244204 A058947 A282912 * A283176 A284296 A283586

Adjacent sequences: A284022 A284023 A284024 * A284026 A284027 A284028

Keyword

nonn,easy

Author

Robert Price, Mar 18 2017

Status

approved