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A068922 Number of ways to tile a 3 X 2n room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

%I A068922 #45 Apr 20 2022 09:44:02
%S A068922 3,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,8362,
%T A068922 13530,21892,35422,57314,92736,150050,242786,392836,635622,1028458,
%U A068922 1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634,78176338,126491972
%N A068922 Number of ways to tile a 3 X 2n room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.
%H A068922 Colin Barker, <a href="/A068922/b068922.txt">Table of n, a(n) for n = 1..1000</a>
%H A068922 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 2.
%H A068922 F. Ruskey and J. Woodcock, <a href="https://doi.org/10.37236/215">Counting Fixed-Height Tatami Tilings</a>, Electronic Journal of Combinatorics, Paper R126 (2009) 20 pages.
%H A068922 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A068922 For n >= 2, a(n) = 2*F(n+1), where F(n)=A000045(n) is the n-th Fibonacci number.
%F A068922 G.f.: x*(x^2-x-3) / (x^2+x-1). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; checked and corrected by _R. J. Mathar_, Sep 16 2009
%F A068922 From _Colin Barker_, Jan 29 2017: (Start)
%F A068922 a(n) = (2^(-n)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / sqrt(5) for n>1.
%F A068922 a(n) = a(n-1) + a(n-2) for n>3. (End)
%F A068922 E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - 2 + x. - _Stefano Spezia_, Apr 18 2022
%p A068922 with(combinat): 3,seq(2*fibonacci(n+1),n=2..40); # _Muniru A Asiru_, Oct 07 2018
%t A068922 Join[{3}, Table[2 Fibonacci[n + 1], {n, 2, 50}]] (* _Vincenzo Librandi_, Oct 07 2018 *)
%t A068922 CoefficientList[Series[(x^2-x-3) / (x^2+x-1), {x, 0, 50}], x] (* _Stefano Spezia_, Oct 07 2018 *)
%o A068922 (PARI) Vec(x*(3+x-x^2) / (1-x-x^2) + O(x^50)) \\ _Colin Barker_, Jan 29 2017
%o A068922 (Magma) [3] cat [2*Fibonacci(n+1): n in [2..50]]; // _Vincenzo Librandi_, Oct 07 2018
%o A068922 (GAP) Concatenation([3],List([2..40],n->2*Fibonacci(n+1))); # _Muniru A Asiru_, Oct 07 2018
%Y A068922 Cf. A068928 for incongruent tilings, A068920 for more info. First column of A272472.
%Y A068922 Essentially the same as A006355.
%Y A068922 Essentially the same as A078642. - _Georg Fischer_, Oct 06 2018
%K A068922 easy,nonn
%O A068922 1,1
%A A068922 _Dean Hickerson_, Mar 11 2002