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A219862 Number of tilings of a 4 X n rectangle using dominoes and straight (3 X 1) trominoes.

%I A219862 #19 Aug 20 2024 09:41:51
%S A219862 1,1,7,41,184,1069,5624,29907,161800,862953,4631107,24832532,
%T A219862 133028028,713283085,3822965706,20491221900,109840081931,588746006676,
%U A219862 3155783700063,16915482096570,90669231898345,486001022349368,2605035346917456,13963368769216664
%N A219862 Number of tilings of a 4 X n rectangle using dominoes and straight (3 X 1) trominoes.
%H A219862 Alois P. Heinz, <a href="/A219862/b219862.txt">Table of n, a(n) for n = 0..1000</a>
%H A219862 <a href="/index/Rec#order_45">Index entries for linear recurrences with constant coefficients</a>, signature (4, 6, 27, -102, 17, -395, 797, -644, 2838, -2657, 2523, -8602, 4310, -8873, 16066, -2769, 18628, -15477, 3863, -27627, 5040, -9300, 19846, 1875, 15731, -6435, 1924, -7786, -680, -4783, 1842, -657, 1108, 32, 734, -278, 88, -32, 10, -58, 13, -3, -1, 0, 1).
%F A219862 G.f.: see Maple program.
%e A219862 a(2) = 7, because there are 7 tilings of a 4 X 2 rectangle using dominoes and straight (3 X 1) trominoes:
%e A219862 .___.   .___.   .___.   .___.   .___.   .___.   .___.
%e A219862 | | |   |___|   |___|   | | |   |___|   |___|   | | |
%e A219862 | | |   | | |   |___|   |_|_|   | | |   |___|   |_|_|
%e A219862 |_|_|   | | |   |___|   |___|   |_|_|   | | |   | | |
%e A219862 |___|   |_|_|   |___|   |___|   |___|   |_|_|   |_|_|
%p A219862 gf:= -(x^42 +x^41 -4*x^40 +4*x^38 -41*x^37 +16*x^36 +45*x^35 +67*x^34 -166*x^33 +282*x^32 -148*x^31 +155*x^30 -405*x^29 +995*x^28 -1118*x^27 +575*x^26 -1863*x^25 +402*x^24 -3552*x^23 +2577*x^22 -406*x^21 +5797*x^20 -741*x^19 +3045*x^18 -5606*x^17 +223*x^16 -4294*x^15 +2924*x^14 -753*x^13 +3011*x^12 -1029*x^11 +811*x^10 -1205*x^9 +248*x^8 -310*x^7 +229*x^6 -17*x^5 +53*x^4 -20*x^3 -3*x^2 -3*x +1) /
%p A219862 (x^45 -x^43 -3*x^42 +13*x^41 -58*x^40 +10*x^39 -32*x^38 +88*x^37 -278*x^36 +734*x^35 +32*x^34 +1108*x^33 -657*x^32 +1842*x^31 -4783*x^30 -680*x^29 -7786*x^28 +1924*x^27 -6435*x^26 +15731*x^25 +1875*x^24 +19846*x^23 -9300*x^22 +5040*x^21 -27627*x^20 +3863*x^19 -15477*x^18 +18628*x^17 -2769*x^16 +16066*x^15 -8873*x^14 +4310*x^13 -8602*x^12 +2523*x^11 -2657*x^10 +2838*x^9 -644*x^8 +797*x^7 -395*x^6 +17*x^5 -102*x^4 +27*x^3 +6*x^2 +4*x -1):
%p A219862 a:= n-> coeff(series(gf, x, n+1), x, n):
%p A219862 seq(a(n), n=0..30);
%Y A219862 Cf. A165716, A165791, A165799, A190759, A202536.
%Y A219862 Column k=4 of A219866.
%K A219862 nonn,easy
%O A219862 0,3
%A A219862 _Alois P. Heinz_, Nov 29 2012