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Number of tilings of an n X 9 rectangle using integer-sided rectangular tiles of area n.
2

%I #19 Sep 05 2021 19:16:52

%S 1,1,55,19,281,6,473,4,495,37,75,1,1091,1,60,30,495,1,609,1,309,22,55,

%T 1,1509,6,55,37,286,1,499,1,495,19,55,9,1259,1,55,19,523,1,478,1,281,

%U 48,55,1,1509,4,75,19,281,1,609,6,500,19,55,1,1125,1,55,40

%N Number of tilings of an n X 9 rectangle using integer-sided rectangular tiles of area n.

%C 1 followed by period 2520: (1, 55, ..., 1716) repeated; offset 0.

%H Alois P. Heinz, <a href="/A220134/b220134.txt">Table of n, a(n) for n = 0..2520</a>

%F G.f.: see Maple program.

%e a(7) = 4, because there are 4 tilings of a 7 X 9 rectangle using integer-sided rectangular tiles of area 7:

%e ._._._._._._._._._. ._____________._._.

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%e ._._____________._. ._._._____________.

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%p gf:= -(1715*x^84 -1714*x^83 +1769*x^82 -35*x^81 +2032*x^80 -255*x^79 +2518*x^78 -443*x^77 +4730*x^76 -427*x^75 +4827*x^74 +13*x^73 +9665*x^72 -2709*x^71 +8592*x^70 +1768*x^69 +11758*x^68 -521*x^67 +13222*x^66 -578*x^65 +17124*x^64 +1707*x^63 +15652*x^62 +761*x^61 +24022*x^60

%p -1869*x^59 +20608*x^58 +3671*x^57 +25352*x^56 +340*x^55 +25138*x^54 +265*x^53 +28854*x^52 +3728*x^51 +25940*x^50 -281*x^49 +33322*x^48 +443*x^47 +26340*x^46 +4094*x^45 +30372*x^44 +578*x^43 +28000*x^42 +578*x^41 +28658*x^40 +4094*x^39 +24626*x^38 -1271*x^37 +28180*x^36

%p +1433*x^35 +20798*x^34 +2014*x^33 +21998*x^32 +265*x^31 +18282*x^30 +340*x^29 +16782*x^28 +1957*x^27 +13752*x^26 -1869*x^25 +13738*x^24 +761*x^23 +8796*x^22 -7*x^21 +8554*x^20 -578*x^19 +6366*x^18 -521*x^17 +4902*x^16 +54*x^15 +3450*x^14 -995*x^13 +2809*x^12 +13*x^11 +1399*x^10 -427*x^9 +1302*x^8 -443*x^7 +804*x^6 -255*x^5 +318*x^4-35*x^3 +55*x^2+1)/

%p (x^84 -x^83 +x^82 +x^80 +x^78 +2*x^76 +2*x^74 +4*x^72 -x^71 +3*x^70 +x^69 +4*x^68 +4*x^66 +5*x^64 +x^63 +4*x^62 +6*x^60 +4*x^58 +x^57 +5*x^56 +4*x^54 +4*x^52 +x^51 +3*x^50 -x^49 +3*x^48 +x^47 +x^46 +x^44 -x^40 -x^38

%p -x^37 -3*x^36 +x^35 -3*x^34 -x^33 -4*x^32 -4*x^30 -5*x^28 -x^27 -4*x^26 -6*x^24 -4*x^22 -x^21 -5*x^20 -4*x^18 -4*x^16 -x^15 -3*x^14 +x^13 -4*x^12 -2*x^10 -2*x^8 -x^6 -x^4 -x^2 +x -1):

%p a:= n-> coeff(series(gf, x, n+1), x, n):

%p seq(a(n), n=0..100);

%Y Row n=9 of A220122.

%K nonn,easy

%O 0,3

%A _Alois P. Heinz_, Dec 06 2012