A080204 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.

1, 3, 10, 23, 51, 114, 253, 559, 1234, 2723, 6007, 13250, 29225, 64459, 142170, 313567, 691595, 1525362, 3364293, 7420183, 16365730, 36095755, 79611695, 175589122, 387274001, 854159699, 1883908522, 4155091047, 9164341795, 20212592114, 44580275277, 98324892351, 216862376818

Offset

1,2

Comments

See Baake-Sing (2002/2003) for the Kolakowski sequence.

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

Links

Metin Sariyar, Table of n, a(n) for n = 1..300

J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated C*-algebras, Ergod. Theory and Dyn. Systems 18 (1998) 509-537. See also, Technical report DMS-720-IR, University of Victoria, 1995.

Michael Baake and B. Sing, Kolakoski-(3,1) is a (deformed) model set, arXiv:math/0206098 [math.MG], 2002-2003.

Greg Dresden and Michael Tulskikh, Tilings of 2 X n boards with dominos and L-shaped trominos, Washington & Lee University (2021).

Index entries for linear recurrences with constant coefficients, signature (3,-2,1,-1).

Formula

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.

From Colin Barker, Jul 02 2012: (Start)

a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) - a(n-4).

G.f.: x*(1+3*x^2-2*x^3)/((1-x)*(1-2*x-x^3)). (End)

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

a(n) = 2*a(n-1) + a(n-3) + 2. - Greg Dresden, Feb 08 2020

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

Mathematica

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 *)

Prog

(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
(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 */
(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

Crossrefs

Equals one less than A332647.

Sequence in context: A256525 A192973 A294503 * A115982 A167243 A316403

Adjacent sequences: A080201 A080202 A080203 * A080205 A080206 A080207

Keyword

easy,nonn

Author

Michael Baake and Uwe Grimm, Mar 20 2003

Status

approved