%I #26 Apr 08 2024 18:52:57

%S 1,1,2,5,10,22,49,105,227,494,1071,2322,5038,10927,23699,51405,111498,

%T 241837,524546,1137742,2467761,5352577,11609747,25181550,54618807,

%U 118468250,256957750,557341615,1208874523,2622050045,5687229162,12335605733,26755941146

%N Number of ways to tile a 2 X n grid with dominoes and L-trominoes such that no four tiles meet at a corner.

%C a(n) <= A052980(n).

%H Peter Kagey, <a href="/A329185/b329185.txt">Table of n, a(n) for n = 0..2500</a>

%H Misha Lavrov, <a href="https://math.stackexchange.com/a/3426264/121988">Number of ways to tile a room with I-Shaped and L-Shaped Tiles</a>, Mathematics Stack Exchange.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,3,-1,2).

%F a(n) = 2*a(n-1) - a(n-2) + 3*a(n-3) - a(n-4) + 2*a(n-5), with a(0) = a(1) = 1, a(2) = 2, a(3) = 5, and a(4) = 10.

%F G.f.: (1 - x)*(1 + x^2) / (1 - 2*x + x^2 - 3*x^3 + x^4 - 2*x^5). - _Colin Barker_, Nov 12 2019

%e For n=3, the five tilings are:

%e +---+---+---+ +---+---+---+

%e | | | | | | |

%e + + + + + +---+---+

%e | | | | | | |

%e +---+---+---+, +---+---+---+,

%e +---+---+---+ +---+---+---+

%e | | | | | |

%e +---+---+ + + +---+ +

%e | | | | | |

%e +---+---+---+, +---+---+---+, and

%e +---+---+---+

%e | | |

%e + +---+ +

%e | | |

%e +---+---+---+.

%e For n=4, the only tiling counted by A052980(4) that is not counted by a(4) is

%e +---+---+---+---+

%e | | |

%e +---+---+---+---+

%e | | |

%e +---+---+---+---+.

%t LinearRecurrence[{2, -1, 3, -1, 2}, {1, 1, 2, 5, 10}, 50] (* _Paolo Xausa_, Apr 08 2024 *)

%o (PARI) Vec((1 - x)*(1 + x^2) / (1 - 2*x + x^2 - 3*x^3 + x^4 - 2*x^5) + O(x^30)) \\ _Colin Barker_, Nov 12 2019

%Y A052980 is the analogous problem without the "four corners" restriction.

%K nonn,easy

%O 0,3

%A _Peter Kagey_, Nov 07 2019