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.

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, 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, 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

Offset: 1

Author

Bradley Klee, Aug 10 2015

Keywords

• nonn
• easy

Comments

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:

P[n] |---> P[n], M[n - 1], M[n], P[n + 1], P[n], P[n], M[n + 1];

M[n] |---> P[n + 1], M[n], M[n], M[n + 1], P[n], P[n - 1], M[n];

with all arithmetic evaluated modulo 3.

The numeric sequence changes the signed vectors M[n] and P[n] into exponent coefficients according to another set of replacement rules:

P[n] |---> Mod[2 n, 6];

M[n] |---> Mod[2 n + 3, 6].

The axiom for sequence is P[0]=0; however, other axioms are just as good.

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 }.

From Michel Dekking, Oct 14 2022: (Start)

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.

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^*.

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.

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)

Links

• Kevin Ryde, Table of n, a(n) for n = 1..10000
• J. H. Conway, Chaim Goodman-Strauss, and N. J. A. Sloane, Recent progress in sphere packing, Current Developments in Mathematics, (1999) 37-76.
• F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
• Martin Gardner, Mathematical Games: In which "monster" curves force redefinition of the word "curve", Scientific American, volume 235, number 6, December 1976, pages 124-133.
• Martin Gardner, Penrose Tiles to Trapdoor Ciphers: And the Return of Dr Matrix, Mathematical Association of America, 1996, chapter 3 (revised and expanded reprint of Mathematical Games article).
• Bradley Klee, A Pit of Flowsnakes, Complex Systems, 24, 4 (2015), section 2.
• Kevin Ryde, PARI/GP Code
• Index entries for sequences that are fixed points of mappings
• Index entries for two-way infinite sequences

Crossrefs

Cf. A229214 (as +-1,2,3), A261185 (mod 2), A261120.

Coordinates: A334485, A334486.

Programs

Mathematica

FLSN = {P[n_] :> {P[n], M[n - 1], M[n], P[n + 1], P[n], P[n], M[n + 1]},
M[n_] :> {P[n + 1], M[n], M[n], M[n + 1], P[n], P[n - 1], M[n]}};
a[1]=P[0];Map[(a[n_/;IntegerQ[(n - #)/7]]:=Part[Flatten[a[(n + 7 - #)/7] /. FLSN], #]) &, Range[7]];
Mod[a /@ Range[7*7]/.{P[x_]:>Mod[2 x, 6],M[x_]:>Mod[2 x + 3, 6]}, 6]

PARI

\\ See links.