A269875 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 41", based on the 5-celled von Neumann neighborhood.

3, 1, 27, -19, 71, -55, 139, -135, 235, -207, 355, -363, 527, -459, 627, -643, 859, -779, 1003, -1007, 1319, -1279, 1563, -1563, 1887, -1803, 2175, -2163, 2495, -2375, 2751, -2767, 3195, -3071, 3607, -3683, 4107, -3931, 4479, -4603, 5191, -4899, 5287, -5299

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..99

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

code=41; stages=100;
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 *)

Crossrefs

Cf. A269872.

Sequence in context: A113099 A317930 A270078 * A271203 A271151 A027495

Adjacent sequences: A269872 A269873 A269874 * A269876 A269877 A269878

Keyword

sign,easy

Author

Robert Price, Mar 06 2016

Status

approved