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A273489
Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 758", based on the 5-celled von Neumann neighborhood.
0
1, 5, 21, 85, 413, 1765, 7325, 30085, 122813, 499045, 2020445, 8156485, 32854013, 132108325, 530526365, 2128417285
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.
FORMULA
Conjecture: a(n) = 2*4^n - 4*3^(n-1) + 2*2^n - 23, n>2. - Lars Blomberg, Jul 20 2016
Conjectures from Colin Barker, Jul 20 2016: (Start)
a(n) = 10*a(n-1)-35*a(n-2)+50*a(n-3)-24*a(n-4) for n>6.
G.f.: (1-5*x+6*x^2+72*x^4-320*x^5+384*x^6) / ((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=758; 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 *)
Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)
CROSSREFS
Cf. A273486.
Sequence in context: A187063 A026855 A272832 * A097113 A368345 A265939
KEYWORD
nonn,more
AUTHOR
Robert Price, May 23 2016
EXTENSIONS
a(8)-a(15) from Lars Blomberg, Jul 20 2016
STATUS
approved