

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.


5



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
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OFFSET

1,3


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*7n)+3) mod 6, and (a(11*7n)+3) mod 6. The valid twoway infinite sequences are generated from M[n]P[m], n != m, or: { 10, 50, 12, 32, 34, 54 }.


LINKS

Table of n, a(n) for n=1..105.
J. H. Conway, Chaim GoodmanStrauss, and N. J. A. Sloane, Recent progress in sphere packing, Current Developments in Mathematics, (1999) 3776.
Martin Gardner, Mathematical Games: In which "monster" curves force redefinition of the word "curve", Scientific American, volume 235, number 6, December 1976, pages 124133.
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.
Index entries for twoway infinite sequences


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]


CROSSREFS

Cf. A229214 (as +1,2,3), A261185 (mod 2), A261120.
Coordinates: A334485, A334486.
Sequence in context: A322114 A062787 A131370 * A062707 A160230 A293500
Adjacent sequences: A261177 A261178 A261179 * A261181 A261182 A261183


KEYWORD

nonn


AUTHOR

Bradley Klee, Aug 10 2015


STATUS

approved



