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

1,1

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).

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

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 *)

Prog

(PARI) Vec(x*(3+x-x^2) / (1-x-x^2) + O(x^50)) \\ Colin Barker, Jan 29 2017
(Magma) [3] cat [2*Fibonacci(n+1): n in [2..50]]; // Vincenzo Librandi, Oct 07 2018
(GAP) Concatenation([3], List([2..40], n->2*Fibonacci(n+1))); # Muniru A Asiru, Oct 07 2018

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

Sequence in context: A214289 A310005 A355325 * A032408 A347567 A018908

Adjacent sequences: A068919 A068920 A068921 * A068923 A068924 A068925

Keyword

easy,nonn

Author

Dean Hickerson, Mar 11 2002

Status

approved