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A261180 Flowsnake phases, exp(I 2 Pi a(n) / 6) are vectors in a sequence that visits points of the hexagonal root lattice A_2.

%I A261180 #58 Dec 19 2024 11:46:19
%S A261180 0,1,3,2,0,0,5,0,1,1,3,4,2,1,2,3,3,5,0,4,3,2,3,5,4,2,2,1,0,1,3,2,0,0,
%T A261180 5,0,1,3,2,0,0,5,4,5,5,1,2,0,5,0,1,3,2,0,0,5,0,1,1,3,4,2,1,0,1,1,3,4,
%U A261180 2,1,2,3,3,5,0,4,3,4,5,1,0,4,4,3,2,3,5,4,2,2,1,0,1,1,3,4,2,1,2,3,5,4,2,2,1
%N A261180 Flowsnake phases, exp(I 2 Pi a(n) / 6) are vectors in a sequence that visits points of the hexagonal root lattice A_2.
%C A261180 This sequence is generated by a Lindenmayer system over six symbols, { M[n], P[n] } with n in {0,1,2}. The replacement rules are:
%C A261180 P[n] |---> P[n], M[n - 1], M[n], P[n + 1], P[n], P[n], M[n + 1];
%C A261180 M[n] |---> P[n + 1], M[n], M[n], M[n + 1], P[n], P[n - 1], M[n];
%C A261180 with all arithmetic evaluated modulo 3.
%C A261180 The numeric sequence changes the signed vectors M[n] and P[n] into exponent coefficients according to another set of replacement rules:
%C A261180 P[n] |---> Mod[2 n, 6];
%C A261180 M[n] |---> Mod[2 n + 3, 6].
%C A261180 The axiom for sequence is P[0]=0; however, other axioms are just as good.
%C A261180 a(n) is one of three right infinite sequences. The other right infinite sequences are a(3*7+n) and a(11*7+n). If n is a negative number, the left infinite sequences are (a(-n)+3) mod 6, (a(-3*7-n)+3) mod 6, and (a(-11*7-n)+3) mod 6. The valid two-way infinite sequences are generated from M[n]|P[m], n != m, or: { 1|0, 5|0, 1|2, 3|2, 3|4, 5|4 }.
%C A261180 From _Michel Dekking_, Oct 14 2022: (Start)
%C A261180 This sequence is a 7-automatic sequence on the alphabet A = {0,1,2,3,4,5}, fixed point with starting letter 0 of a morphism alpha.
%C A261180 Let sigma be the rotation on A given by sigma(a) = a+1 mod 6, and let rho be the reversal map given by rho(w_1...w_m) = w_m...w_1 for all words w_1...w_m in A^*.
%C A261180 The morphism alpha is defined by alpha(0) = 0132005, and by requiring that alpha commutes with the map sigma rho. So, for example, alpha(1) = 0113421.
%C A261180 See A229214 for another form of (a(n)). The standard form of (a(n)) is given by the sequence x = 1,2,3,4,1,1,5,1,2,2,3,6,4,2,...(First map A to {1,...,6} by a->a+1, and then apply the permutation (34)(56)). (End)
%H A261180 Kevin Ryde, <a href="/A261180/b261180.txt">Table of n, a(n) for n = 1..10000</a>
%H A261180 J. H. Conway, Chaim Goodman-Strauss, and N. J. A. Sloane, <a href="http://math.uark.edu/Recent_Progress_in_Sphere_Packing.pdf"> Recent progress in sphere packing</a>, Current Developments in Mathematics, (1999) 37-76.
%H A261180 F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%H A261180 Martin Gardner, <a href="http://www.jstor.org/stable/24950510">Mathematical Games: In which "monster" curves force redefinition of the word "curve"</a>, Scientific American, volume 235, number 6, December 1976, pages 124-133.
%H A261180 Martin Gardner, <a href="http://www.maa.org/sites/default/files/pdf/pubs/focus/Gardner_Fractals12-1976.pdf">Penrose Tiles to Trapdoor Ciphers: And the Return of Dr Matrix</a>, Mathematical Association of America, 1996, chapter 3 (revised and expanded reprint of Mathematical Games article).
%H A261180 Bradley Klee, <a href="http://www.complex-systems.com/abstracts/v24_i04_a01.html">A Pit of Flowsnakes</a>, Complex Systems, 24, 4 (2015), section 2.
%H A261180 Kevin Ryde, <a href="/A261180/a261180.gp.txt">PARI/GP Code</a>
%H A261180 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%H A261180 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%t A261180 FLSN = {P[n_] :> {P[n], M[n - 1], M[n], P[n + 1], P[n], P[n], M[n + 1]},
%t A261180 M[n_] :> {P[n + 1], M[n], M[n], M[n + 1], P[n], P[n - 1], M[n]}};
%t A261180 a[1]=P[0];Map[(a[n_/;IntegerQ[(n - #)/7]]:=Part[Flatten[a[(n + 7 - #)/7] /. FLSN], #]) &, Range[7]];
%t A261180 Mod[a /@ Range[7*7]/.{P[x_]:>Mod[2 x, 6],M[x_]:>Mod[2 x + 3, 6]}, 6]
%o A261180 (PARI) \\ See links.
%Y A261180 Cf. A229214 (as +-1,2,3), A261185 (mod 2), A261120.
%Y A261180 Coordinates: A334485, A334486.
%K A261180 nonn,easy
%O A261180 1,3
%A A261180 _Bradley Klee_, Aug 10 2015