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A322663 First differences of A322662 divided by 12. 1
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, 39, 70, 10, 42, 41, 81, 4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, 18, 37, 64, 151, 41, 80, 126, 131, 29 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..56.

FORMULA

a(n) = (A322662(n)-A322662(n-1))/12.

EXAMPLE

Written as a 2^k triangle:

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, 39, 70,  10, 42, 41, 81,

4, 11, 21, 50, 24, 57, 74, 89, 40, 62, 84, 105, 48, 66, 85, 111, ...

MATHEMATICA

HexStar=2*Sqrt[3]*{Cos[#*Pi/3+Pi/6], Sin[#*Pi/3+Pi/6]}&/@Range[0, 5];

MoveSet2 =Join[2*HexStar+RotateRight[HexStar], 2*HexStar+RotateLeft[HexStar]];

Clear@Pts; Pts[0] = {{0, 0}};

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]]];

Abs[(1/12)*Subtract@@#&/@Partition[Length[Pts[#]]&/@Range[0, 32], 2, 1]]

CROSSREFS

Hexagonal: A151724, A170898, A256537. Square: A147582, A147610, A048883; A319019, A322050, A322049. Lower Bound: A038573.

Sequence in context: A019796 A309600 A190264 * A231927 A199076 A132728

Adjacent sequences:  A322660 A322661 A322662 * A322664 A322665 A322666

KEYWORD

nonn

AUTHOR

Bradley Klee, Dec 22 2018

STATUS

approved

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Last modified December 10 04:15 EST 2019. Contains 329885 sequences. (Running on oeis4.)