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A229214 If 1, 2, and 3 represent the three 2D vectors (1,0), (0.5,sqrt(3)/2) and (-0.5,sqrt(3)/2) and -1, -2 and -3 are the negation of these vectors, then this sequence represents the Gosper flowsnake. 6

%I

%S 1,2,-1,3,1,1,-3,1,2,2,-1,-2,3,2,3,-1,-1,-3,1,-2,-1,3,-1,-3,-2,3,3,2,

%T 1,2,-1,3,1,1,-3,1,2,-1,3,1,1,-3,-2,-3,-3,2,3,1,-3,1,2,-1,3,1,1,-3,1,

%U 2,2,-1,-2,3,2,1,2,2,-1,-2,3,2,3,-1,-1,-3,1,-2,-1,-2,-3,2,1,-2,-2,-1

%N If 1, 2, and 3 represent the three 2D vectors (1,0), (0.5,sqrt(3)/2) and (-0.5,sqrt(3)/2) and -1, -2 and -3 are the negation of these vectors, then this sequence represents the Gosper flowsnake.

%C The sequence is generated by the rewriting rules:

%C P(1) = 1,2,-1,3,1,1,-3;

%C P(2) = 1,2,2,-1,-2,3,2 and

%C P(3) = 3,-1,-3,-2,3,3,2;

%C P(-x) = reverse(-P(x)) for x=1,2,3, so

%C P(-1) = 3,-1,-1,-3,1,-2,-1,

%C P(-2) = -2,-3,2,1,-2,-2,-1, and

%C P(-3) = -2,-3,-3,2,3,1,-3.

%C The start is 1.

%H Arie Bos, <a href="http://arxiv.org/abs/1210.7123">Index notation of grid graphs</a>, arXiv:1210.7123 [cs.CG], 2012.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Gosper_curve">Gosper curve</a>

%e Start with 1, you get in the first step 1, 2, -1, 3, 1, 1, -3,

%e and in the 2nd step 1, 2, -1, 3, 1, 1, -3, 1, 2, 2, -1, -2, 3, 2, 3, -1, -1, -3, 1, -2, -1, 3, -1, -3, -2, 3, 3, 2, 1, 2, -1, 3, 1, 1, -3, 1, 2, -1, 3, 1, 1, -3, -2, -3, -3, 2, 3, 1, -3

%e and with each step the length increases by a factor 7.

%o (PARI) A229214(n,P=[[1,2,-1,3,1,1,-3],[1,2,2,-1,-2,3,2],[3,-1,-3,-2,3,3,2]],a=P[1])={while(#a<n,a=concat(apply(i->if(i<0,-Vecrev(P[-i]),P[i]),a)));a} \\ _M. F. Hasler_, Aug 06 2015

%Y Cf. A261180 (as 0..5). Coordinates: A334485, A334486.

%Y Cf. A229215 (Gosper island directions).

%K easy,sign

%O 1,2

%A _Arie Bos_, Sep 19 2013

%E Definition corrected by _Kerry Mitchell_, Aug 06 2015

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Last modified April 20 18:45 EDT 2021. Contains 343137 sequences. (Running on oeis4.)