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.

3, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972

Offset: 1

Author

Dean Hickerson, Mar 11 2002

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 2.
• F. Ruskey and J. Woodcock, Counting Fixed-Height Tatami Tilings, Electronic Journal of Combinatorics, Paper R126 (2009) 20 pages.
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A068928 for incongruent tilings, A068920 for more info. First column of A272472.

Essentially the same as A006355.

Essentially the same as A078642. - Georg Fischer, Oct 06 2018

Programs

GAP

Concatenation([3],List([2..40],n->2*Fibonacci(n+1))); # Muniru A Asiru, Oct 07 2018

Magma

[3] cat [2*Fibonacci(n+1): n in [2..50]]; // Vincenzo Librandi, Oct 07 2018

Maple

with(combinat): 3,seq(2*fibonacci(n+1),n=2..40); # Muniru A Asiru, Oct 07 2018

Mathematica

Join[{3}, Table[2 Fibonacci[n + 1], {n, 2, 50}]] (* Vincenzo Librandi, Oct 07 2018 *)
CoefficientList[Series[(x^2-x-3) / (x^2+x-1), {x, 0, 50}], x] (* Stefano Spezia, Oct 07 2018 *)

PARI

Vec(x*(3+x-x^2) / (1-x-x^2) + O(x^50)) \\ Colin Barker, Jan 29 2017

Formula

For n >= 2, a(n) = 2*F(n+1), where F(n)=A000045(n) is the n-th Fibonacci number.

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

From Colin Barker, Jan 29 2017: (Start)

a(n) = (2^(-n)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / sqrt(5) for n>1.

a(n) = a(n-1) + a(n-2) for n>3. (End)

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