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%I #11 Dec 07 2012 12:53:08
%S 1,0,2,4,8,28,65,170,456,1177,3068,8016,21028,54696,142908,374527,
%T 976022,2551162,6674999,17422250,45536000,119053392,310969076,
%U 812695944,2124175469,5550025712,14503987368,37905500955,99050326532,258846292750,676453686574
%N Number of tilings of a 5 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
%H Alois P. Heinz, <a href="/A219969/b219969.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: see Maple program.
%e a(4) = 8, because there are 8 tilings of a 5 X 4 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
%e ._____._. ._._____. ._._._._. ._._.___.
%e |_____| | | |_____| | | | | | | | | |
%e |_____| | | |_____| | | | | | | | |___|
%e |_____|_| |_|_____| |_|_|_|_| |_|_| | |
%e | | | | | | | | | | | | |
%e |___|___| |___|___| |___|___| |___|_|_|
%e .___.___. .___.___. .___.___. .___._._.
%e | | | | | | | | | | | | |
%e |___|___| |___|___| |___|___| |___| | |
%e |_____| | | |_____| | | | | | | | |_|_|
%e |_____| | | |_____| | | | | | | | | |
%e |_____|_| |_|_____| |_|_|_|_| |_|_|___|
%p gf:= -(x^81 -7*x^78 +41*x^75 +x^73 -198*x^72 +2*x^71 -10*x^70 +845*x^69 -16*x^68 +43*x^67 -3156*x^66 +86*x^65 -96*x^64 +10444*x^63 -331*x^62 +68*x^61 -30704*x^60 +991*x^59 +335*x^58 +80592*x^57 -2465*x^56 -1564*x^55 -189222*x^54 +5338*x^53 +3968*x^52 +397848*x^51 -10648*x^50 -7680*x^49 -747706*x^48 +20835*x^47 +13544*x^46 +1251990*x^45 -40621*x^44 -24871*x^43 -1858564*x^42 +74789*x^41 +47191*x^40 +2433033*x^39
%p -121355*x^38 -82950*x^37 -2791787*x^36 +165741*x^35 +123957*x^34 +2789367*x^33 -185980*x^32 -151345*x^31 -2407340*x^30 +169318*x^29 +148399*x^28 +1776356*x^27 -123893*x^26 -115498*x^25 -1105831*x^24 +71944*x^23 +70340*x^22 +570573*x^21 -32495*x^20 -32842*x^19 -238424*x^18 +11077*x^17 +11417*x^16 +78374*x^15 -2727*x^14 -2832*x^13 -19542*x^12 +453*x^11 +469*x^10 +3523*x^9 -45*x^8 -46*x^7 -428*x^6 +2*x^5 +2*x^4 +31*x^3 -1)*(x -1)^2*(x^2 +x +1)^2 /
%p (x^90 +2*x^88 -9*x^87 +6*x^86 -26*x^85 +63*x^84 -76*x^83 +183*x^82 -367*x^81 +546*x^80 -954*x^79 +1830*x^78 -2884*x^77 +3929*x^76 -7765*x^75 +12072*x^74 -13027*x^73 +28518*x^72 -41491*x^71 +35304*x^70 -91935*x^69 +119871*x^68 -78938*x^67 +262994*x^66 -296401*x^65 +145610*x^64 -672074*x^63 +635235*x^62 -216634*x^61 +1540902*x^60 -1188099*x^59 +240723*x^58 -3175160*x^57 +1942824*x^56 -139170*x^55 +5876128*x^54 -2771239*x^53 -161593*x^52 -9748106*x^51 +3426351*x^50 +683453*x^49
%p +14467189*x^48 -3628004*x^47 -1368932*x^46 -19177263*x^45 +3210694*x^44 +2087516*x^43 +22669949*x^42 -2240035*x^41 -2665512*x^40 -23841863*x^39 +1009987*x^38 +2932037*x^37 +22213131*x^36 +87664*x^35 -2788225*x^34 -18207162*x^33 -751490*x^32 +2278157*x^31 +13000933*x^30 +911285*x^29 -1585085*x^28 -7987552*x^27 -725314*x^26 +928121*x^25 +4159353*x^24 +433937*x^23 -448956*x^22 -1802433*x^21 -202164*x^20 +174393*x^19 +635455*x^18 +73626*x^17 -52289*x^16 -177158*x^15 -20603*x^14 +11476*x^13 +37649*x^12 +4257*x^11 -1709*x^10 -5802*x^9 -605*x^8 +152*x^7 +602*x^6 +52*x^5 -6*x^4 -37*x^3 -2*x^2 +1):
%p a:= n-> coeff (series (gf, x, n+1), x, n):
%p seq (a(n), n=0..50);
%Y Column k=5 of A219967.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Dec 02 2012
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