A273744 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 901", based on the 5-celled von Neumann neighborhood.

1, 9, 34, 83, 164, 285, 454, 679, 968, 1329, 1770, 2299, 2924, 3653, 4494, 5455, 6544, 7769, 9138, 10659, 12340, 14189, 16214, 18423, 20824, 23425, 26234, 29259, 32508, 35989, 39710, 43679, 47904, 52393, 57154, 62195, 67524, 73149, 79078, 85319, 91880, 98769

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) = n*(11+12*n+4*n^2)/3 for n>0.

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

G.f.: (1+5*x+4*x^2-3*x^3+x^4) / (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=901; 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. A273743.

Sequence in context: A262959 A139275 A236370 * A133547 A100179 A106598

Adjacent sequences: A273741 A273742 A273743 * A273745 A273746 A273747

Keyword

nonn,easy

Author

Robert Price, May 28 2016

Status

approved