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
KEYWORD
easy,nonn
AUTHOR
Michael Baake and Uwe Grimm, Mar 20 2003
STATUS
approved