A273316 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood.

1, 9, 33, 81, 161, 281, 449, 673, 961, 1321, 1761, 2289, 2913, 3641, 4481, 5441, 6529, 7753, 9121, 10641, 12321, 14169, 16193, 18401, 20801, 23401, 26209, 29233, 32481, 35961, 39681, 43649, 47873, 52361, 57121, 62161, 67489, 73113, 79041, 85281, 91841, 98729

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

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, May 19 2016: (Start)

a(n) = (4*n^3+12*n^2+8*n+3)/3.

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3.

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

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=645; 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[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)

Crossrefs

Cf. A033996.

Sequence in context: A205796 A081585 A227221 * A101990 A147170 A146823

Adjacent sequences: A273313 A273314 A273315 * A273317 A273318 A273319

Keyword

nonn,easy

Author

Robert Price, May 19 2016

Status

approved