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A269512
First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 342", based on the 5-celled von Neumann neighborhood.
1
4, 3, 16, -7, 36, -21, 64, -47, 100, -77, 144, -119, 196, -165, 256, -223, 324, -285, 400, -359, 484, -437, 576, -527, 676, -621, 784, -727, 900, -837, 1024, -959, 1156, -1085, 1296, -1223, 1444, -1365, 1600, -1519, 1764, -1677, 1936, -1847, 2116, -2021
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, Apr 03 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.: (4+7*x+15*x^2+2*x^3+6*x^4-x^5-x^6) / ((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=342; 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. A264797.
Sequence in context: A065679 A223925 A062776 * A270128 A084471 A285122
KEYWORD
sign,easy
AUTHOR
Robert Price, Apr 03 2016
STATUS
approved