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A229215
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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.
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3
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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
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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