A397812 Number of tilings of a 3 X n rectangle, with all four corners removed, using dominoes and right trominoes.

0, 1, 0, 3, 4, 27, 68, 315, 980, 3883, 13188, 49355, 173716, 635483, 2268676, 8224667, 29529044, 106669131, 383840260, 1384587819, 4986809492, 17978203387, 64774493572, 233469116027, 841296459988, 3032043498091, 10926465128068, 39377714668171, 141907245404308

Offset

1,4

Comments

Compare to A165716 which counts the same type of tiling on a 3 X n rectangle, but including the four corners.

Links

Table of n, a(n) for n=1..29.

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

Formula

a(n) = 2*a(n-1) + 6*a(n-2) - 4*a(n-3) + 11*a(n-4) + 2*a(n-5) for n >= 7.

G.f.: x^2*(1 - 2*x - 3*x^2 + 2*x^3 - 10*x^4)/(1 - 2*x - 6*x^2 + 4*x^3 - 11*x^4 - 2*x^5). - Stefano Spezia, Jul 11 2026

Example

a(4) = 3, because there are 3 tilings of a 3 X 4 rectangle (missing the four corners) using dominoes and right trominoes:
     ___        ___        ___
   _| | |_    _|___|_    _|___|_
  |___|___|  |_  |  _|  |___|___|
    |___|      |_|_|      |___|

Mathematica

CoefficientList[Series[x^2*(1 - 2*x - 3*x^2 + 2*x^3 - 10*x^4)/(1 - 2*x - 6*x^2 + 4*x^3 - 11*x^4 - 2*x^5), {x, 0, 29}], x] (* Stefano Spezia, Jul 11 2026 *)

Crossrefs

Cf. A165716.

Sequence in context: A157163 A042225 A094084 * A362887 A367388 A042829

Adjacent sequences: A397802 A397803 A397808 * A397813 A397816 A397818

Keyword

nonn,easy,new

Author

Greg Dresden and Jinpeng (Parker) Ou, Jul 10 2026

Status

approved