A284485 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 961", based on the 5-celled von Neumann neighborhood.

1, 0, 7, 11, 15, 47, 63, 191, 255, 767, 1023, 3071, 4095, 12287, 16383, 49151, 65535, 196607, 262143, 786431, 1048575, 3145727, 4194303, 12582911, 16777215, 50331647, 67108863, 201326591, 268435455, 805306367, 1073741823, 3221225471, 4294967295, 12884901887

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

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

Robert Price, Diagrams of first 20 stages

Formula

Conjectures from Colin Barker, Mar 28 2017: (Start)

G.f.: (1 - x + 3*x^2 + 8*x^3 - 24*x^4 + 16*x^5) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).

a(n) = (-4 - (-2)^n + 5*2^n)/4 for n>2.

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>5.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 961; 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. A284483, A284484, A101622.

Sequence in context: A097494 A037136 A023486 * A089997 A129188 A022950

Adjacent sequences: A284482 A284483 A284484 * A284486 A284487 A284488

Keyword

nonn,easy

Author

Robert Price, Mar 27 2017

Status

approved