A283587 Decimal 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 654", based on the 5-celled von Neumann neighborhood.

1, 3, 7, 13, 29, 53, 113, 219, 475, 859, 1819, 3515, 7595, 13739, 29099, 56235, 121771, 220075, 465835, 900011, 1944491, 3517355, 7449515, 14396331, 31173547, 56339371, 119253931, 230402987, 497789867, 900443051, 1907076011, 3685460907, 7980428203

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 12 2017: (Start)

G.f.: (1 + 2*x + 4*x^2 + 6*x^3 + 16*x^4 + 24*x^5 + 60*x^6 + 106*x^7 - 128*x^9 - 64*x^10 + 160*x^11 - 16*x^12) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 + 4*x^2)*(1 + 16*x^4)).

a(n) = a(n-1) + 256*a(n-8) - 256*a(n-9) for n>12.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 654; 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]], 2], {i, 1, stages - 1}]

Crossrefs

Cf. A283585, A283586, A283588.

Sequence in context: A080166 A116872 A161490 * A283705 A284401 A284542

Adjacent sequences: A283584 A283585 A283586 * A283588 A283589 A283590

Keyword

nonn,easy

Author

Robert Price, Mar 11 2017

Status

approved