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A273147
First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 597", based on the 5-celled von Neumann neighborhood.
1
7, 12, 21, 16, 47, 12, 77, 0, 119, -20, 165, -48, 223, -84, 285, -128, 359, -180, 437, -240, 527, -308, 621, -384, 727, -468, 837, -560, 959, -660, 1085, -768, 1223, -884, 1365, -1008, 1519, -1140, 1677, -1280, 1847, -1428, 2021, -1584, 2207, -1748, 2397
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.
FORMULA
Conjectures from Colin Barker, May 16 2016: (Start)
a(n) = -a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-5)-a(n-6)-a(n-7) for n>6.
G.f.: (7+19*x+26*x^2+18*x^3+23*x^4+3*x^5) / ((1-x)^2*(1+x)^3*(1+x^2)).
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=597; 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. A273144.
Sequence in context: A025006 A189321 A273072 * A272845 A272927 A272946
KEYWORD
sign,easy
AUTHOR
Robert Price, May 16 2016
STATUS
approved