A290685 Binary 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 934", based on the 5-celled von Neumann neighborhood.

1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 101100000, 1011000000, 10110000000, 101100000000, 1011000000000, 10110000000000, 101100000000000, 1011000000000000, 11110001000000000, 111100010000000000, 1111000100000000000, 11110001000000000000

Offset

0,2

Comments

Initialized with a single black (ON) cell at stage zero.

Curiously, for many values of n (e.g. 9<=n<=23), a(n) satisfies the linear recurrence 11*a(n) - 110*a(n-1) - 10000010000*a(n-8) + 100000100000*a(n-9). For instance, out of the first 127 terms, 76 terms satisfy this recurrence. - Chai Wah Wu, Aug 03 2020

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

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 934; 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. A290684, A290686, A290687.

Sequence in context: A290237 A286832 A290623 * A290853 A290849 A290668

Adjacent sequences: A290682 A290683 A290684 * A290686 A290687 A290688

Keyword

nonn,easy

Author

Robert Price, Aug 09 2017

Status

approved