A106796 - OEIS

%I #24 Apr 05 2022 03:26:32

%S 1,1,2,1,1,2,3,1,1,2,1,1,2,3,1,4,1,1,2,1,1,2,3,1,1,2,1,1,2,3,1,4,1,1,

%T 2,1,1,1,2,1,1,2,3,1,1,2,1,1,2,3,1,4,1,1,2,1,1,2,3,1,1,2,1,1,2,3,1,4,

%U 1,1,2,1,1,1,2,1,1,2,3,1,1,2,1,1,2,1,1,2,3,1,1,2,1,1,2,3,1,4,1,1,2,1,1,2,3

%N Fixed point of the morphism  1 -> 1,1,2; 2 -> 3;  3 -> 1,4; 4 -> 1, starting with a(0) = 1.

%C 4-symbol substitution for the Pisot characteristic polynomial: x^4 - 2*x^2 - x - 1.

%H G. C. Greubel, <a href="/A106796/b106796.txt">Table of n, a(n) for n = 0..10000</a>

%H Victor F. Sirvent and Boris Solomyak, <a href="https://doi.org/10.4153/CMB-2002-062-3">Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type</a>. Canadian Mathematical Bulletin, 45(4), 2002, 697-710; (page 709 example 3). Also at <a href="https://www.researchgate.net/publication/228561314_Pure_Discrete_Spectrum_for_One-dimensional_Substitution_Systems_of_Pisot_Type">ResearchGate</a>

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%e The first few steps of the substitution are:

%e Start: 1

%e Maps:

%e   1 --> 1 1 2

%e   2 --> 3

%e   3 --> 1 4

%e   4 --> 1

%e -------------

%e 0:   (#=1)

%e   1

%e 1:   (#=3)

%e   112

%e 2:   (#=7)

%e   1121123

%e 3:   (#=16)

%e   1121123112112314

%e 4:   (#=36)

%e   112112311211231411211231121123141121

%e 5:   (#=82)

%e   1121123112112314112112311211231411211121123112112314112112311211231411211121123112

%t s[1]= {1, 1, 2}; s[2]= {3}; s[3]= {1, 4}; s[4]= {1}; t[b_]:= Flatten[s /@ b];

%t a[0]= {1}; a[1]= t[p[0]]; a[n_]:= t[a[n-1]];

%t a[10]

%Y Cf. A106749, A106795, A106797, A106798.

%K nonn

%O 0,3

%A _Roger L. Bagula_, May 17 2005

%E Edited by _G. C. Greubel_, Apr 03 2022