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A272586
Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 507", based on the 5-celled von Neumann neighborhood.
0
1, 5, 40, 200, 872, 3624, 14760, 59560, 239272, 959144, 3840680, 15370920, 61500072, 246033064, 984197800, 3936922280
OFFSET
0,2
COMMENTS
Initialized with a single black (ON) cell at stage zero.
Similar to A271286 (Rule 505). Only a(1) appears to be different through 8 terms.
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
FORMULA
Conjecture: a(n) = (11*4^n-12*2^n-8)/3 , n>1. - Lars Blomberg, Jul 08 2016
Conjectures from Colin Barker, Jul 08 2016: (Start)
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4.
G.f.: (1-2*x+19*x^2-18*x^3-8*x^4) / ((1-x)*(1-2*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=507; 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. A272585.
Perhaps this is related to A271286? - Alois P. Heinz, May 03 2016
Sequence in context: A272226 A271278 A271290 * A271084 A271092 A273653
KEYWORD
nonn,more
AUTHOR
Robert Price, May 03 2016
EXTENSIONS
a(8)-a(15) from Lars Blomberg, Jul 08 2016
STATUS
approved