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%I #13 Feb 06 2019 13:07:36
%S 1,1,7,1,6,11,14,3,11,14,25,5,18,21,37,4,11,21,50,17,31,50,50,13,32,
%T 39,70,10,42,41,81,4,11,21,50,24,57,74,89,40,62,84,105,48,66,85,111,
%U 18,37,64,151,41,80,126,131,29
%N First differences of A322662 divided by 12.
%C Unlike A322050, this sequence contains only finitely many 1's. However, the Cellular Automaton and its counting sequences still admit a 2^n fractal structure (Cf. A322662). The subsequences L_n = {a(2^n), a(2^n+1), ... a(2^(n+1)-1)} appear to approach a limit sequence L_{oo}, starting with 4 ON cells. Of these 4, one is a "pioneer" at distance d*2^n from the origin, with d the distance of one knight step. The other three of four ON cells are due to retrogressive growth.
%F a(n) = (A322662(n)-A322662(n-1))/12.
%e Written as a 2^k triangle:
%e 1,
%e 1, 7,
%e 1, 6, 11, 14,
%e 3, 11, 14, 25, 5, 18, 21, 37,
%e 4, 11, 21, 50, 17, 31, 50, 50, 13, 32, 39, 70, 10, 42, 41, 81,
%e 4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, ...
%t HexStar=2*Sqrt[3]*{Cos[#*Pi/3+Pi/6],Sin[#*Pi/3+Pi/6]}&/@Range[0,5];
%t MoveSet2 =Join[2*HexStar+RotateRight[HexStar],2*HexStar+RotateLeft[HexStar]];
%t Clear@Pts;Pts[0] = {{0, 0}};
%t Pts[n_]:=Pts[n]=With[{pts=Pts[n-1]},Union[pts,Cases[Tally[Flatten[pts/.{x_,y_}:> Evaluate[{x,y}+#&/@MoveSet2],1]],{x_,1}:>x]]];
%t Abs[(1/12)*Subtract@@#&/@Partition[Length[Pts[#]]&/@Range[0,32],2,1]]
%Y Hexagonal: A151724, A170898, A256537. Square: A147582, A147610, A048883; A319019, A322050, A322049. Lower Bound: A038573.
%K nonn
%O 1,3
%A _Bradley Klee_, Dec 22 2018
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