A190759 Number of tilings of a 5 X n rectangle using right trominoes and 2 X 2 tiles.

1, 0, 4, 0, 16, 0, 136, 0, 1128, 384, 8120, 6912, 60904, 75136, 491960, 720640, 4023592, 6828928, 32819320, 63472640, 270471784, 574543744, 2256221368, 5119155712, 18940876712, 45266369152, 159625747960, 397949457408, 1350573713256

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..650

Formula

G.f.: (20*x^12+40*x^11 +18*x^10+52*x^9 +35*x^8-26*x^7 +34*x^6-4*x^5 -21*x^4 +2*x^3 +9*x^2-1) / (-80*x^14-160*x^13 +308*x^12-88*x^11 -2*x^10+396*x^9 -453*x^8-10*x^7 +190*x^6-12*x^5 -57*x^4+2*x^3 +13*x^2-1).

Example

a(2) = 4, because there are 4 tilings of a 5 X 2 rectangle using right trominoes and 2 X 2 tiles:
.___. .___. .___. .___.
| . | | . | | ._| |_. |
|___| |___| |_| | | |_|
| ._| |_. | |___| |___|
|_| | | |_| | . | | . |
|___| |___| |___| |___|

Maple

a:= n-> (Matrix(14, (i, j)-> `if`(i=j-1, 1, `if`(i=14, [-80, -160, 308, -88, -2, 396, -453, -10, 190, -12, -57, 2, 13, 0][j], 0)))^n. <<0, 1/4, 0, 1, 0, 4, 0, 16, 0, 136, 0, 1128, 384, 8120>>)[4, 1]: seq(a(n), n=0..30);

Mathematica

a[n_] := (MatrixPower[ Table[ If[i == j-1, 1, If[i == 14, {-80, -160, 308, -88, -2, 396, -453, -10, 190, -12, -57, 2, 13, 0}[[j]], 0]], {i, 1, 14}, {j, 1, 14}], n] . {0, 1/4, 0, 1, 0, 4, 0, 16, 0, 136, 0, 1128, 384, 8120})[[4]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 05 2013, translated from Alois P. Heinz's Maple program *)

Crossrefs

Cf. A165799, A165791, A165716, A054854, A054856.

Column k=5 of A219946.

Sequence in context: A210067 A199572 A003195 * A086262 A167361 A321610

Adjacent sequences: A190756 A190757 A190758 * A190760 A190761 A190762

Keyword

easy,nice,nonn

Author

Alois P. Heinz, May 18 2011

Status

approved