A272708 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 515", based on the 5-celled von Neumann neighborhood.

4, 8, 12, 24, 20, 40, 28, 56, 36, 72, 44, 88, 52, 104, 60, 120, 68, 136, 76, 152, 84, 168, 92, 184, 100, 200, 108, 216, 116, 232, 124, 248, 132, 264, 140, 280, 148, 296, 156, 312, 164, 328, 172, 344, 180, 360, 188, 376, 196, 392, 204, 408, 212, 424, 220, 440

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

Stephen 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

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Formula

From Mike Sheppard, Feb 04 2025: (Start)

a(n) = 2*a(n-2) - a(n-4).

a(n) = 2*(1 + 3*n) + (-1)^n*(2 - 2*n).

a(2*n) = 4+8*n.

a(2*n-1) = -8+16*n.

G.f.: 4 * (1 + 2*x + x^2 + 2*x^3) / (-1 + x^2)^2. (End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=515; 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 *)
LinearRecurrence[{0, 2, 0, -1}, {4, 8, 12, 24}, 128] (* Mike Sheppard, Feb 04 2025 *)

Crossrefs

Cf. A272705.

Sequence in context: A130643 A014617 A239053 * A157416 A278602 A059992

Adjacent sequences: A272705 A272706 A272707 * A272709 A272710 A272711

Keyword

nonn,easy

Author

Robert Price, May 04 2016

Status

approved