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A165716 Number of tilings of a 3 X n rectangle using dominoes and right trominoes. 8
1, 0, 5, 8, 55, 140, 633, 1984, 7827, 26676, 99621, 351080, 1283247, 4583580, 16611505, 59652624, 215457835, 775371268, 2796772765, 10073343672, 36315180295, 130843331180, 471599612393, 1699398816608, 6124635653443, 22071172760532, 79541846573973 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..550

Index entries for linear recurrences with constant coefficients, signature (2,6,-4,11,2).

FORMULA

G.f.: (2*x^4 - 2*x^3 + x^2 + 2*x - 1) / (2*x^5 + 11*x^4 - 4*x^3 + 6*x^2 + 2*x - 1).

a(0)=1, a(1)=0, a(2)=5, a(3)=8, a(4)=55, a(n) = 2*a(n-1) + 6*a(n-2) - 4*a(n-3) + 11*a(n-4) + 2*a(n-5). - Harvey P. Dale, Mar 19 2013

EXAMPLE

a(2) = 5, because there are 5 tilings of a 3 X 2 rectangle using dominoes and right trominoes:

.___. .___. ._._. .___. .___.

|___| |_._| | | | | ._| |_. |

|___| | | | |_|_| |_| | | |_|

|___| |_|_| |___| |___| |___|

MAPLE

a:= n-> (Matrix([[55, 8, 5, 0, 1]]). Matrix(5, (i, j)-> if i=j-1 then 1 elif j=1 then [2, 6, -4, 11, 2][i] else 0 fi)^n)[1, 5]: seq(a(n), n=0..25);

MATHEMATICA

a[n_] := Last[{55, 8, 5, 0, 1} . MatrixPower[ Table[ Which[i == j - 1, 1, j == 1, {2, 6, -4, 11, 2}[[i]], True, 0], {i, 1, 5}, {j, 1, 5}], n]]; Table[a[n], {n, 0, 24}] (* Jean-Fran├žois Alcover, Jul 19 2012, translated from Maple *)

LinearRecurrence[{2, 6, -4, 11, 2}, {1, 0, 5, 8, 55}, 30] (* Harvey P. Dale, Mar 19 2013 *)

CROSSREFS

Column k=3 of A219987.

Sequence in context: A294665 A323139 A284381 * A068478 A342195 A132050

Adjacent sequences:  A165713 A165714 A165715 * A165717 A165718 A165719

KEYWORD

easy,nice,nonn,changed

AUTHOR

Alois P. Heinz, Sep 24 2009

STATUS

approved

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Last modified November 28 22:15 EST 2021. Contains 349415 sequences. (Running on oeis4.)