A229215 - OEIS

%I #28 Jun 12 2024 09:40:44

%S 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,

%T -3,1,-3,-2,-3,-2,-1,-2,-3,-2,-3,-2,-1,-2,-1,3,-1,-2,-1,-2,-3,-2,-3,

%U -2,-1,-2,-3,-2,-3,1,-3,1,-3,-2,-3,1,-3,1,-3,-2,-3,-2

%N 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.

%C The sequence is generated by the rewriting rules

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

%C P(2) = 2,1,2,

%C P(3) = 3,2,3,

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

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

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

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

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

%C 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

%H Paolo Xausa, <a href="/A229215/b229215.txt">Table of n, a(n) for n = 1..19683</a>

%H Arie Bos, <a href="http://arxiv.org/abs/1210.7123">Index notation of grid graphs</a>, arXiv:1210.7123 [cs.CG], 2012.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Gosper_curve">Gosper curve</a>

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%e Start with 1,2,3,-1,-2,-3 and you get

%e in the first step 1,-3,1,2,1,2,3,2,3,-1,3,-1,-2,-1,-2,-3,-2,-3 and

%e in the second step 1,-3,1,-3,-2,-3,1,-3,1,2,1,2,1,-3, ... ,-1,-2,-3,-2,-3.

%e With each step the length increases by a factor of 3.

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

%o (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

%Y Cf. A229214.

%K easy,sign

%O 1,2

%A _Arie Bos_, Sep 24 2013

%E Definition corrected by _Kerry Mitchell_, Aug 06 2015