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%I #78 Jun 15 2022 01:51:21
%S 1,3,10,23,51,114,253,559,1234,2723,6007,13250,29225,64459,142170,
%T 313567,691595,1525362,3364293,7420183,16365730,36095755,79611695,
%U 175589122,387274001,854159699,1883908522,4155091047,9164341795,20212592114,44580275277,98324892351,216862376818
%N Number of fixed points under n-fold inflation for the substitution rule a->abc, b->ab, c->b that underlies the Kolakoski (3,1) sequence.
%C See Baake-Sing (2002/2003) for the Kolakowski sequence.
%C a(n) is the number of possible tilings of a bracelet of "thickness" 1 and length n using single-color squares, single-color dominoes, and two-color k-ominoes with k >= 3. - _Michael Tulskikh_ and _Greg Dresden_, Sep 03 2019; edited by _Greg Dresden_, Feb 18 2020, May 14 2020, May 18 2020
%H Metin Sariyar, <a href="/A080204/b080204.txt">Table of n, a(n) for n = 1..300</a>
%H J. E. Anderson and I. F. Putnam, <a href="https://doi.org/10.1017/S0143385798100457">Topological invariants for substitution tilings and their associated C*-algebras</a>, Ergod. Theory and Dyn. Systems 18 (1998) 509-537. See <a href="http://hdl.handle.net/1828/2786">also</a>, Technical report DMS-720-IR, University of Victoria, 1995.
%H Michael Baake and B. Sing, <a href="http://arxiv.org/abs/math/0206098">Kolakoski-(3,1) is a (deformed) model set</a>, arXiv:math/0206098 [math.MG], 2002-2003.
%H Greg Dresden and Michael Tulskikh, <a href="https://dresden.academic.wlu.edu/files/2021/01/TwoByNWeb.pdf">Tilings of 2 X n boards with dominos and L-shaped trominos</a>, Washington & Lee University (2021).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1,-1).
%F Obtained from dynamical zeta function Z(x)=(1-x)/(1-2x-x^3), so that x Z'(x)/Z(x) is the ordinary power series generating function.
%F From _Colin Barker_, Jul 02 2012: (Start)
%F a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) - a(n-4).
%F G.f.: x*(1+3*x^2-2*x^3)/((1-x)*(1-2*x-x^3)). (End)
%F a(n) = n*Sum_{m=1..n} Sum_{i=0..(n-m)/2} binomial(m,i)*binomial(n-2*i-1,m-1)/m. - _Vladimir Kruchinin_, Mar 10 2015
%F a(n) = 2*a(n-1) + a(n-3) + 2. - _Greg Dresden_, Feb 08 2020
%F a(n) = w1^n + w2^n + w3^n - 1, for w1, w2, w3 roots of x^3-2x^2-1=0. - _Greg Dresden_, Feb 18 2020
%t CoefficientList[Series[(1 + 3 x^2 - 2 x^3) / ((1 - x) (1 - 2 x - x^3)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 12 2015 *)
%o (PARI) x='x+O('x^66); Vec(x*(1+3*x^2-2*x^3)/((1-x)*(1-2*x-x^3))) \\ _Joerg Arndt_, Jun 15 2013
%o (Maxima) a(n):=(n*sum(sum(binomial(m,i)*binomial(n-2*i-1,m-1),i,0,(n-m)/2)/m,m,1,n)); /* _Vladimir Kruchinin_, Mar 10 2015 */
%o (Magma) I:=[1,3,10,23]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)+Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Mar 12 2015
%Y Equals one less than A332647.
%K easy,nonn
%O 1,2
%A Michael Baake and Uwe Grimm, Mar 20 2003
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