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.

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, 1, 2, -1, 3, 1, 1, -3, 1, 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

Offset: 1

Author

Arie Bos, Sep 19 2013

Keywords

• easy
• sign

Comments

The sequence is generated by the rewriting rules:

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

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

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

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

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

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

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

The start is 1.

Examples

			Start with 1, you get in the first step 1, 2, -1, 3, 1, 1, -3,
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
and with each step the length increases by a factor 7.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..16807
• Arie Bos, Index notation of grid graphs, arXiv:1210.7123 [cs.CG], 2012.
• Wikipedia, Gosper curve

Crossrefs

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

Cf. A229215 (Gosper island directions).

Programs

Mathematica

With[{p = {{1,2,-1,3,1,1,-3}, {1,2,2,-1,-2,3,2}, {3,-1,-3,-2,3,3,2}}}, SubstitutionSystem[{t_/; t > 0 :> p[[t]], t_ :> -Reverse[p[[-t]]]}, {1}, {3}][[1]]] (* Paolo Xausa, Jun 12 2024 *)

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(#aif(i<0,-Vecrev(P[-i]),P[i]),a)));a} \\ M. F. Hasler, Aug 06 2015

Extensions

Definition corrected by Kerry Mitchell, Aug 06 2015