A068921 Number of ways to tile a 2 X n room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.
1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
- Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 10.
- Richard J. Mathar, Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 1.
- Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 7.
- Index entries for linear recurrences with constant coefficients, signature (1,0,1).
Crossrefs
Programs
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Mathematica
LinearRecurrence[{1, 0, 1}, {1, 1, 2}, 42] (* Robert G. Wilson v, Jul 12 2014 *) -
PARI
my(x='x+O('x^50)); Vec((1+x^2)/(1-x-x^3)) \\ G. C. Greubel, Apr 26 2017
Formula
For n >= 3, a(n) = a(n-1) + a(n-3).
a(n) = A000930(n+1).
From Frank Ruskey, Jun 07 2009: (Start)
G.f.: (1+x^2)/(1-x-x^3).
a(n) = Sum_{j=0..floor(n/2)} binomial(n-2j+1, j). (End)
G.f.: Q(0)*( 1+x^2 )/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + x^2)/( x*(4*k+3 + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 08 2013