login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A165799
Number of tilings of a 4 X n rectangle using right trominoes and 2 X 2 tiles.
6
1, 0, 1, 4, 6, 16, 37, 92, 245, 560, 1426, 3720, 9069, 22808, 58177, 145660, 366318, 925536, 2331269, 5872212, 14802941, 37311528, 94038250, 236999064, 597348237, 1505640016, 3794761257, 9564393972, 24106951622, 60759989040, 153141435269, 385986293964
OFFSET
0,4
FORMULA
G.f.: -(6*x^3+x-1) / (4*x^9+16*x^7+22*x^6+3*x^5-x^4-9*x^3-x^2-x+1).
a(n) = a(n-1) +a(n-2) +9*a(n-3) +a(n-4) -3*a(n-5) -22*a(n-6) -16*a(n-7) -4*a(n-9).
EXAMPLE
a(4) = 6, because there are 6 tilings of a 4 X 4 rectangle using right trominoes and 2 X 2 tiles:
.___.___. .___.___. .___.___. .___.___. .___.___. .___.___.
| . | . | | ._|_. | | ._| . | | ._|_. | | ._|_. | | . |_. |
|___|___| |_| . |_| |_| |___| |_| ._|_| |_|_. |_| |___| |_|
| . | . | | |___| | | |___| | | |_| . | | . |_| | | |___| |
|___|___| |___|___| |___|___| |___|___| |___|___| |___|___|
MAPLE
a:= n-> (Matrix([[4, 1, 0, 1, 0$5]]). Matrix(9, (i, j)-> if i=j-1 then 1 elif j=1 then [1, 1, 9, 1, -3, -22, -16, 0, -4][i] else 0 fi)^n)[1, 4]: seq(a(n), n=0..30);
MATHEMATICA
Series[ (-6*x^3 - x + 1) / (4*x^9 + 16*x^7 + 22*x^6 + 3*x^5 - x^4 - 9*x^3 - x^2 - x + 1), {x, 0, 31}] // CoefficientList[#, x] & (* Jean-François Alcover, Jun 18 2013, after Alois P. Heinz *)
LinearRecurrence[{1, 1, 9, 1, -3, -22, -16, 0, -4}, {1, 0, 1, 4, 6, 16, 37, 92, 245}, 40] (* Harvey P. Dale, Nov 09 2024 *)
CROSSREFS
Column k=4 of A219946.
Sequence in context: A188466 A076066 A227178 * A231998 A056421 A032295
KEYWORD
easy,nice,nonn
AUTHOR
Alois P. Heinz, Sep 27 2009
STATUS
approved