A353965 Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles and right trominoes.

1, 1, 13, 47, 259, 1189, 5877, 28167, 136723, 660173, 3194613, 15445007, 74699811, 361230229, 1746933205, 8448061879, 40854753875, 197572345789, 955455626773, 4620559362303, 22344915889827, 108059470995013, 522573007884725, 2527150465444071, 12221238828079379

Offset

0,3

Comments

For tiling algorithm see A351322.

Links

Table of n, a(n) for n=0..24.

Index entries for linear recurrences with constant coefficients, signature (3,9,-1,2).

Formula

G.f.: (1 - 2*x + x^2) / (1 - 3*x - 9*x^2 + x^3 - 2*x^4).

a(n) = 3*a(n-1) + 9*a(n-2) - a(n-3) + 2*a(n-4).

31*a(n) = 18*(-2)^n +13*A200739(n+3) +2*A200739(n+2) +9*A200739(n+1). - R. J. Mathar, Jun 07 2025

Example

a(2) = 13:
    v    h,v   h=v   h,v
   ___   ___   ___   ___   ___
  |   | | |_| |  _| |  _| |_|_|    mirroring included
  |___| |___| |_| | |_|_| |_|_|    h: horizontal, v: vertical
  |_|_| |_|_| |___| |_|_| |_|_|
    2  +  4  +  2  +  4  +  1 = 13

Prog

(Maxima) See A352589.

Crossrefs

Cf. A351322, A352589, A353963, A353964.

Sequence in context: A358287 A146806 A300337 * A355961 A135712 A225920

Adjacent sequences: A353962 A353963 A353964 * A353966 A353967 A353968

Keyword

nonn,easy

Author

Gerhard Kirchner, May 13 2022

Status

approved