A271256 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 323", based on the 5-celled von Neumann neighborhood.

4, 0, 31, -27, 87, -79, 171, -167, 291, -287, 443, -439, 627, -623, 843, -839, 1091, -1087, 1371, -1367, 1683, -1679, 2027, -2023, 2403, -2399, 2811, -2807, 3251, -3247, 3723, -3719, 4227, -4223, 4763, -4759, 5331, -5327, 5931, -5927, 6563, -6559, 7227

Offset

0,1

Comments

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

Similar to A270086.

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

Formula

Conjectures from Colin Barker, Apr 03 2016: (Start)

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

a(n) = 4*n^2+4*n+3 for n>4 and even.

a(n) = -4*n^2+4*n+1 for n>4 and odd.

a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>7.

G.f.: (4+4*x+23*x^2-4*x^3+2*x^4+4*x^5+3*x^6-8*x^7+4*x^9) / ((1-x)^2*(1+x)^3).

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=323; 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 *)

Crossrefs

Cf. A270086, A270569.

Sequence in context: A283917 A284186 A283068 * A271057 A269879 A269817

Adjacent sequences: A271253 A271254 A271255 * A271257 A271258 A271259

Keyword

sign,easy

Author

Robert Price, Apr 02 2016

Status

approved