A229215 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 Gosper's island.

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

Offset: 1

Author

Arie Bos, Sep 24 2013

Keywords

• easy
• sign

Comments

The sequence is generated by the rewriting rules

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

P(2) = 2,1,2,

P(3) = 3,2,3,

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

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

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

The start is 1,2,3,-1,-2,-3.

Notice P(-x)= -P(x), since P(x) is symmetric.

Among the starting values, only the initial "1" is relevant for computation of the sequence, the image of the other elements (2,3,-1,-2,-3) becomes "pushed away" to infinity. - M. F. Hasler, Aug 06 2015

Examples

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

Links

• Paolo Xausa, Table of n, a(n) for n = 1..19683
• Arie Bos, Index notation of grid graphs, arXiv:1210.7123 [cs.CG], 2012.
• Wikipedia, Gosper curve
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A229214.

Programs

Mathematica

SubstitutionSystem[{t_ :> {{1,-3,1}, {2,1,2}, {3,2,3}}[[Abs[t]]]*Sign[t]}, {1}, {3}][[1]] (* Paolo Xausa, Jun 12 2024 *)

PARI

(P(v)=concat(apply(i->[i,i-sign(i)*4^(i*i<2),i],v)));A229215=P(P(P(P([1])))) \\ To get a(n), ceil(log_3(n)) iterations are required. - M. F. Hasler, Aug 06 2015

Extensions

Definition corrected by Kerry Mitchell, Aug 06 2015