login
A270209
First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 121", based on the 5-celled von Neumann neighborhood.
1
3, 5, 23, -15, 83, -60, 124, -123, 259, -211, 351, -364, 560, -503, 667, -627, 915, -895, 1095, -1023, 1335, -1252, 1532, -1507, 1951, -1980, 2305, -2085, 2421, -2421, 2941, -2865, 3313, -3197, 3593, -3633, 4337, -4337, 4781, -4689, 5341, -5169, 5665, -5533
OFFSET
0,1
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=121; 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[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)
CROSSREFS
Cf. A270206.
Sequence in context: A272426 A270180 A270793 * A271464 A088121 A144103
KEYWORD
sign,easy
AUTHOR
Robert Price, Mar 13 2016
STATUS
approved