login
A273394
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 670", based on the 5-celled von Neumann neighborhood.
1
1, 6, 19, 40, 73, 122, 183, 264, 369, 486, 635, 792, 1017, 1270, 1595, 1888, 2337, 2782, 3327, 3828, 4501, 5090, 5867, 6544, 7425, 8366, 9383, 10308, 11517, 12610, 13955, 15192, 16737, 18358, 20031, 21796, 23781, 25666, 27923, 30016, 32425, 34926, 37447
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.
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=670; 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. A273392.
Sequence in context: A096957 A272811 A273206 * A273455 A273571 A273778
KEYWORD
nonn,easy
AUTHOR
Robert Price, May 21 2016
STATUS
approved