A229215 -id:A229215 - cp's OEIS frontend

A229214 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 the Gosper flowsnake.

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

Offset: 1

Arie Bos, Sep 19 2013

The sequence is generated by the rewriting rules:
P(1) = 1,2,-1,3,1,1,-3;
P(2) = 1,2,2,-1,-2,3,2 and
P(3) = 3,-1,-3,-2,3,3,2;
P(-x) = reverse(-P(x)) for x=1,2,3, so
P(-1) = 3,-1,-1,-3,1,-2,-1,
P(-2) = -2,-3,2,1,-2,-2,-1, and
P(-3) = -2,-3,-3,2,3,1,-3.
The start is 1.

			Start with 1, you get in the first step 1, 2, -1, 3, 1, 1, -3,
and in the 2nd step 1, 2, -1, 3, 1, 1, -3, 1, 2, 2, -1, -2, 3, 2, 3, -1, -1, -3, 1, -2, -1, 3, -1, -3, -2, 3, 3, 2, 1, 2, -1, 3, 1, 1, -3, 1, 2, -1, 3, 1, 1, -3, -2, -3, -3, 2, 3, 1, -3
and with each step the length increases by a factor 7.

Paolo Xausa, Table of n, a(n) for n = 1..16807
Arie Bos, Index notation of grid graphs, arXiv:1210.7123 [cs.CG], 2012.
Wikipedia, Gosper curve

Cf. A261180 (as 0..5). Coordinates: A334485, A334486.

Cf. A229215 (Gosper island directions).

With[{p = {{1,2,-1,3,1,1,-3}, {1,2,2,-1,-2,3,2}, {3,-1,-3,-2,3,3,2}}}, SubstitutionSystem[{t_/; t > 0 :> p[[t]], t_ :> -Reverse[p[[-t]]]}, {1}, {3}][[1]]] (* Paolo Xausa, Jun 12 2024 *)

A229214(n,P=[[1,2,-1,3,1,1,-3],[1,2,2,-1,-2,3,2],[3,-1,-3,-2,3,3,2]],a=P[1])={while(#aif(i<0,-Vecrev(P[-i]),P[i]),a)));a} \\ M. F. Hasler, Aug 06 2015

Definition corrected by Kerry Mitchell, Aug 06 2015

A106824 Trajectory of 1 under the morphism 1->13, 2->13223, 3->1323.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, May 20 2005

Only from the 13th term on, this differs from the limit sequence of { 1 -> 131, 2 -> 212, 3 -> 323 } = absolute values of A229215. - M. F. Hasler, Aug 06 2015

J. M. Dumont and A. Thomas, Digital sum problems and substitutions on a finite alphabet, J. Number Theory, 39 (1991), 351-366.
Index entries for sequences that are fixed points of mappings

Cf. A229215.

S:={1=[1,3],2=[1,3,2,2,3],3=[1,3,2,3]}:subs(S,1):subs(S,%):subs(S,%):subs(S,%):subs(S,%); # all brackets have to be removed. - Emeric Deutsch, simplified by M. F. Hasler, Aug 06 2015
S:={1=(1,3),2=(1,3,2,2,3),3=(1,3,2,3)}: (curry(subs,S)@@6)([1]); # Robert Israel, Aug 06 2015

Nest[ Flatten[ # /. {1 -> {1, 3}, 2 -> {1, 3, 2, 2, 3}, 3 -> {1, 3, 2, 3}}] &, {1}, 5] (* Robert G. Wilson v, Jun 20 2005 *)

A106824(n,a=[1],S=[[1,3],[1,3,2,2,3],[1,3,2,3]])={while(#aS[i],a)));a} \\ M. F. Hasler, Aug 06 2015

More terms from Emeric Deutsch, May 30 2005

cp's OEIS Frontend

A229214 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 the Gosper flowsnake.

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