%I #11 Apr 28 2023 20:15:39
%S 1,1,19,55,472,2023,13249,66325,392299,2088856,11877025,64803157,
%T 362823607,1998759703,11123273896,61509329983,341492705365,
%U 1891193243713,10489893539203,58127214942544,322296397820593,1786338231961609,9903234373856059,54893955008138983
%N Number of tilings of a 4 X n rectangle using dominos and 2 X 2 right triangles.
%C Triangles only occur as pairs forming 2 X 2 squares. For program code and additional details, see A362297.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,18,-48,-42,99).
%F a(n) = 4*a(n-1) + 18*a(n-2) - 48*a(n-3) - 42*a(n-4) + 99*a(n-5).
%F G.f.: (9*x^3-3*x^2-3*x+1)/(-99*x^5+42*x^4+48*x^3-18*x^2-4*x+1).
%e a(2) = 19.
%e Partitions of a 2 X 2 square (triangles or dominos):
%e ___ ___ ___ ___
%e | /| |\ | |___| | | |
%e |/__| |__\| |___| |_|_|
%e 2t 2d
%e ___ ___ ___ ___ ___ ___ _ ___ _ _______
%e |2t |2t | |2t |2d | |2d |2t | | |2t | | |only d |
%e |___|___| |___|___| |___|___| |_|___|_| |_______|
%e 4 ways + 4 ways + 4 ways + 2 ways + 5 ways = 19 ways
%e Only dominos: A005178(3) = 5.
%t LinearRecurrence[{4,18,-48,-42,99},{1,1,19,55,472},24] (* _Stefano Spezia_, Apr 20 2023 *)
%Y Column k=2 of A362297.
%Y Cf. A351322, A352432, A352433, A006130, A362299.
%K nonn,easy
%O 0,3
%A _Gerhard Kirchner_, Apr 19 2023