

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.


3



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
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OFFSET

1,2


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


LINKS

Table of n, a(n) for n=1..67.
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


EXAMPLE

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.


PROG

(PARI) (P(v)=concat(apply(i>[i, isign(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


CROSSREFS

Cf. A229214.
Sequence in context: A055189 A106824 A317203 * A123508 A117621 A178055
Adjacent sequences: A229212 A229213 A229214 * A229216 A229217 A229218


KEYWORD

easy,sign


AUTHOR

Arie Bos, Sep 24 2013


EXTENSIONS

Definition corrected by Kerry Mitchell, Aug 06 2015


STATUS

approved



