A273833 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood.

1, 5, 26, 71, 147, 264, 428, 649, 933, 1290, 1726, 2251, 2871, 3596, 4432, 5389, 6473, 7694, 9058, 10575, 12251, 14096, 16116, 18321, 20717, 23314, 26118, 29139, 32383, 35860, 39576, 43541, 47761, 52246, 57002, 62039, 67363, 72984, 78908, 85145, 91701, 98586

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, Jun 01 2016: (Start)

a(n) = (16*n^3+48*n^2-10*n-3*((-1)^n-5))/12 for n>1.

a(n) = (8*n^3+24*n^2-5*n+6)/6 for n>1 and even.

a(n) = (8*n^3+24*n^2-5*n+9)/6 for n>1 and odd.

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

G.f.: (1+2*x+13*x^2+5*x^3-7*x^4+3*x^5-x^6) / ((1-x)^4*(1+x)).

(End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=961; 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. A273831.

Sequence in context: A273419 A273447 A273406 * A273849 A273781 A048395

Adjacent sequences: A273830 A273831 A273832 * A273834 A273835 A273836

Keyword

nonn,easy

Author

Robert Price, May 31 2016

Status

approved