A273768 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 913", based on the 5-celled von Neumann neighborhood.

1, 5, 22, 70, 151, 272, 441, 666, 955, 1316, 1757, 2286, 2911, 3640, 4481, 5442, 6531, 7756, 9125, 10646, 12327, 14176, 16201, 18410, 20811, 23412, 26221, 29246, 32495, 35976, 39697, 43666, 47891, 52380, 57141, 62182, 67511, 73136, 79065, 85306, 91867, 98756

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 29 2016: (Start)

a(n) = (4*n^3+12*n^2+11*n-39)/3 for n>2.

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

G.f.: (1+x+8*x^2+8*x^3-16*x^4+5*x^5+x^6) / (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=913; 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. A273766.

Sequence in context: A222632 A366081 A273336 * A032168 A246211 A000345

Adjacent sequences: A273765 A273766 A273767 * A273769 A273770 A273771

Keyword

nonn,easy

Author

Robert Price, May 29 2016

Status

approved