A284543 Decimal 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 974", based on the 5-celled von Neumann neighborhood.

1, 3, 7, 11, 23, 43, 71, 219, 503, 1003, 1991, 4059, 8183, 16363, 32711, 65499, 131063, 262123, 524231, 1048539, 2097143, 4194283, 8388551, 16777179, 33554423, 67108843, 134217671, 268435419, 536870903, 1073741803, 2147483591, 4294967259, 8589934583

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 Colin Barker, Mar 29 2017: (Start)

G.f.: (1 + x + x^2 - 3*x^3 - 4*x^5 - 16*x^6 + 80*x^7 + 64*x^8) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + x^2)).

a(n) = 2*a(n-1) + a(n-4) - 2*a(n-5) for n>8.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 974; 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]], 2], {i , 1, stages - 1}]

Crossrefs

Cf. A284540, A284541, A284542.

Sequence in context: A283588 A283706 A284402 * A267052 A267009 A116606

Adjacent sequences: A284540 A284541 A284542 * A284544 A284545 A284546

Keyword

nonn,easy

Author

Robert Price, Mar 28 2017

Status

approved