A270455 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 163", based on the 5-celled von Neumann neighborhood.

1, 6, 15, 48, 73, 158, 207, 368, 449, 710, 831, 1216, 1385, 1918, 2143, 2848, 3137, 4038, 4399, 5520, 5961, 7326, 7855, 9488, 10113, 12038, 12767, 15008, 15849, 18430, 19391, 22336, 23425, 26758, 27983, 31728, 33097, 37278, 38799, 43440, 45121, 50246, 52095

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, Mar 17 2016: (Start)

a(n) = (3*(3+(-1)^n)+(22-6*(-1)^n)*n-6*(-4+(-1)^n)*n^2+8*n^3)/12.

a(n) = (4*n^3+9*n^2+8*n+6)/6 for n even.

a(n) = (4*n^3+15*n^2+14*n+3)/6 for n odd.

a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>6.

G.f.: (1+5*x+6*x^2+18*x^3+x^4+x^5) / ((1-x)^4*(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=163; 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. A270454.

Sequence in context: A271537 A271891 A272449 * A318414 A106272 A056423

Adjacent sequences: A270452 A270453 A270454 * A270456 A270457 A270458

Keyword

nonn,easy

Author

Robert Price, Mar 17 2016

Status

approved