A106035 The "Octanacci" sequence: Trajectory of 1 under the morphism 1->{1,2,1}, 2->{1}.

1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2

Offset: 0

Author

Roger L. Bagula, May 05 2005

Keywords

• nonn

Comments

Silver mean chain substitution sequence: characteristic polynomial = -x^2+2*x+1.

A space-filling lattice is given by: bb = aa /. 1 -> {-0.4142135623730951, 2.414213562373095} /. 2 -> {1,-0.414213562373095`} /. 3 -> 0; ListPlot[FoldList[Plus, {0, 0}, bb], PlotRange -> All, PlotJoined -> False, Axes -> False];

The sequence is S_oo where S_0 = 2, S_1 = 1; S_{n+2} = S_{n+1} S_n S_{n+1}. Used to construct the "labyrinth" tiling. - N. J. A. Sloane, Mar 13 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1392
• M. Baake and R. V. Moody, Self-Similar Measures for Quasicrystals, in Directions in Mathematical Quasicrystals (eds. M. Baake and R. V. Moody), CRM Monograph Series, vol. 13, AMS, Providence, RI (2000), pp. 1-42; arXiv:math/0008063 [math.MG], 2000.
• Clément Sire, Rémy Mosseri, and Jean-François Sadoc, Geometric study of a 2D tiling related to the octagonal quasiperiodic tiling, Journal de Physique 50.24 (1989): 3463-3476. See Eq. 2; HAL Id : jpa-00211156.
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A085301, A106036.

See A324772 for version over {0,1}.

Programs

Maple

f(1):= (1, 2, 1): f(2):= (1): A:= [1]:
for i from 1 to 6 do A:= map(f, A) od:
A; # - N. J. A. Sloane, Mar 13 2019

Mathematica

s[1] = {1, 2, 1}; s[2] = {1}; s[3] = {}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]] aa = p[6]
Nest[Function[l, Flatten[l/.{1->{1, 2, 1}, 2->{1}}]], {1}, 6] (* Vincenzo Librandi, Mar 14 2019 *)
SubstitutionSystem[{1->{1,2,1},2->{1}},{1},{6}]//Flatten (* Harvey P. Dale, Nov 20 2021 *)