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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 (list; graph; refs; listen; history; text; internal format)
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

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, 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

CROSSREFS

Cf. A229214.

Sequence in context: A016571 A055189 A106824 * 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

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Last modified October 22 11:25 EDT 2017. Contains 293761 sequences.