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Number of ways to tile a 3-row trapezoid of average length n with triangular and rectangular tiles, each of size 3.
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%I #18 Sep 12 2024 15:09:49

%S 0,1,3,6,16,43,107,271,695,1769,4499,11464,29202,74360,189382,482339,

%T 1228417,3128538,7967848,20292665,51681683,131623881,335222157,

%U 853749843,2174345679,5537663440,14103422412,35918853816,91478793556,232979863477,593357374127

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

%C Here is the 3-row trapezoid of average 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-row trapezoid

%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.: x*(1 + x)/((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}, {0, 1, 3, 6, 16, 43}, 40]

%Y Cf. A077939, A077961, A375821.

%K nonn,easy

%O 0,3

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