A270336 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 158", based on the 5-celled von Neumann neighborhood.

4, 7, 8, 12, 0, 32, -12, 52, -44, 92, -44, 92, -24, 92, -32, 92, -68, 192, -160, 228, -132, 268, -176, 304, -268, 392, -260, 448, -372, 480, -288, 324, -240, 452, -364, 444, -216, 476, -424, 608, -340, 572, -476, 548, -72, 84, -20, 340, -84, 356, -452, 604

Offset

0,1

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..127

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

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=158; 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}];
  on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
  Table[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)
Differences[Total[#, 2] & /@ CellularAutomaton[{158, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, {20}]] (* JungHwan Min, Mar 16 2016 *)

Crossrefs

Cf. A270333.

Sequence in context: A000606 A215458 A061932 * A270941 A270082 A189223

Adjacent sequences: A270333 A270334 A270335 * A270337 A270338 A270339

Keyword

sign,easy

Author

Robert Price, Mar 15 2016

Status

approved