A290829 Decimal representation of the diagonal from the origin to the corner 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, 1, 5, 15, 7, 63, 31, 255, 127, 1023, 511, 4095, 2047, 16383, 8191, 65535, 32767, 262143, 131071, 1048575, 524287, 4194303, 2097151, 16777215, 8388607, 67108863, 33554431, 268435455, 134217727, 1073741823, 536870911, 4294967295, 2147483647, 17179869183

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 Colin Barker, Aug 12 2017: (Start)

G.f.: (1 + 10*x^3 - 24*x^4 + 16*x^5) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).

a(n) = 2^(n-1) - 1 for n>2 and even.

a(n) = 2^(n+1) - 1 for n>2 and odd.

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]], 10], {i, 1, stages - 1}]

Crossrefs

Cf. A290813, A290827, A290828.

Sequence in context: A161202 A154353 A077348 * A290837 A302841 A113259

Adjacent sequences: A290826 A290827 A290828 * A290830 A290831 A290832

Keyword

nonn,easy

Author

Robert Price, Aug 11 2017

Status

approved