A272705 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.

1, 5, 13, 25, 49, 69, 109, 137, 193, 229, 301, 345, 433, 485, 589, 649, 769, 837, 973, 1049, 1201, 1285, 1453, 1545, 1729, 1829, 2029, 2137, 2353, 2469, 2701, 2825, 3073, 3205, 3469, 3609, 3889, 4037, 4333, 4489, 4801, 4965, 5293, 5465, 5809, 5989, 6349

Offset

0,2

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..128

Robert Price, Diagrams of the 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

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 (1,2,-2,-1,1).

Formula

From Mike Sheppard, Feb 01 2025: (Start)

a(n) = (5/2 + k (-1 + 3 k)) + (-1)^n (-(3/2) + k).

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

a(2n) = 1 + 12 n^2.

a(2n-1) = 9 - 16 n + 12 n^2.

G.f.: (-1 - 4 x - 6 x^2 - 4 x^3 - 9 x^4)/((-1 + x)^3 (1 + x)^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}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 5, 13, 25, 49}, 129] (* Mike Sheppard, Feb 01 2025 *)

Crossrefs

Sequence in context: A147166 A146192 A147265 * A147272 A147427 A147408

Adjacent sequences: A272702 A272703 A272704 * A272706 A272707 A272708

Keyword

nonn,easy

Author

Robert Price, May 04 2016

Status

approved