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%I #47 Oct 12 2022 08:58:10
%S 1,1,2,3,4,6,9,13,19,28,41,60,88,129,189,277,406,595,872,1278,1873,
%T 2745,4023,5896,8641,12664,18560,27201,39865,58425,85626,125491,
%U 183916,269542,395033,578949,848491,1243524,1822473,2670964,3914488,5736961
%N Number of ways to tile a 2 X n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.
%H G. C. Greubel, <a href="/A068921/b068921.txt">Table of n, a(n) for n = 0..1000</a>
%H Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).
%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, Example 10.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1311.6135">Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings</a>, arXiv:1311.6135 [math.CO], 2013, Table 1.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1).
%F For n >= 3, a(n) = a(n-1) + a(n-3).
%F a(n) = A000930(n+1).
%F From _Frank Ruskey_, Jun 07 2009: (Start)
%F G.f.: (1+x^2)/(1-x-x^3).
%F a(n) = sum( binomial(n-2j+1,j), j=0..floor(n/2) ). (End)
%F 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
%t LinearRecurrence[{1, 0, 1}, {1, 1, 2}, 42] (* _Robert G. Wilson v_, Jul 12 2014 *)
%o (PARI) my(x='x+O('x^50)); Vec((1+x^2)/(1-x-x^3)) \\ _G. C. Greubel_, Apr 26 2017
%Y Cf. A068927 for incongruent tilings, A068920 for more info.
%Y Cf. A000930, A078012, first column of A272471.
%K nonn,easy
%O 0,3
%A _Dean Hickerson_, Mar 11 2002
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