%I #20 Sep 12 2024 11:30:38

%S 1,1,2,7,17,41,107,274,693,1766,4504,11465,29194,74364,189391,482327,

%T 1228412,3128559,7967841,20292639,51681711,131623900,335222103,

%U 853749852,2174345752,5537663377,14103422348,35918853952,91478793557,232979863277,593357374262

%N Number of ways to tile a 3-row parallelogram of length n with triangular and rectangular tiles, each of size 3.

%C Here is the 3-row parallelogram of length 6 (with 18 cells):

%C ___ ___ ___ ___ ___ ___

%C | | | | | | |

%C _|___|___|___|___|_ _|___|

%C | | | | | | |

%C _|___|___|___|___|_ _|___|

%C | | | | | | |

%C |___|___|___|___|___|___|,

%C and here are the two types of (triangular and rectangular) tiles of size 3, which can be rotated as needed:

%C ___

%C | |

%C _|___|_ ___________

%C | | | | | | |

%C |___|___|, |___|___|___|.

%C As an example, here is one of the a(6) = 107 ways to tile the 3 x 6 parallelogram:

%C ___ _______ ___________

%C | | | |

%C _| _|_ _|___________|

%C | | | | |

%C _| _| |___|___________|

%C | | | |

%C |___|_______|___________|.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,4,-1,0,-1).

%F a(n) = 2*a(n-1) + 4*a(n-3) - a(n-4) - a(n-6).

%F G.f.: (1 - x - x^3)/((1 + x^2 - x^3)*(1 - 2*x - x^2 - x^3)).

%F a(n) = (A077939(n) + A077961(n))/2.

%t LinearRecurrence[{2, 0, 4, -1, 0, -1}, {1, 1, 2, 7, 17, 41}, 40]

%Y Cf. A077939, A077961, A375823.

%K nonn,easy

%O 0,3

%A _Greg Dresden_ and Mingjun Oliver Ouyang, Aug 30 2024