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A106035
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The "Octanacci" sequence: Trajectory of 1 under the morphism 1->{1,2,1}, 2->{1}.
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3
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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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.
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MAPLE
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f(1):= (1, 2, 1): f(2):= (1): A:= [1]:
for i from 1 to 6 do A:= map(f, A) od:
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MATHEMATICA
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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 *)
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CROSSREFS
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See A324772 for version over {0,1}.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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