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Number of tilings of a 5 X n rectangle using n pentominoes of shapes T, W, Y, X.
2

%I #123 May 06 2024 17:55:50

%S 1,0,0,0,2,2,8,0,22,40,134,100,266,448,1610,2454,4806,7064,19774,

%T 38320,81174,128604,277540,553762,1222204,2210510,4352240,8339138,

%U 17869740,34938578,69204722,131277114,267512514,533554754,1074570418,2076822340,4120024394

%N Number of tilings of a 5 X n rectangle using n pentominoes of shapes T, W, Y, X.

%H Alois P. Heinz, <a href="/A366324/b366324.txt">Table of n, a(n) for n = 0..3367</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentomino">Pentomino</a>

%H <a href="/index/Rec#order_44">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 0, 5, 9, 6, -5, 5, 8, 48, -13, -37, -36, 58, 37, -37, -94, 8, 56, 188, -254, -181, 92, -55, 155, -267, -346, 91, 444, -61, 56, -258, 36, 371, 116, -122, -94, 40, 116, 64, 0, 16, 0, 32).

%F G.f.: (16*x^42 +32*x^41 +16*x^40 +20*x^39 +8*x^38 +4*x^37 -40*x^36 +8*x^35 +13*x^34 -42*x^33 -84*x^32 +4*x^31 -79*x^30 +82*x^29 +61*x^28 -50*x^27 -33*x^26 +47*x^25 +21*x^24 +36*x^23 -23*x^22 -102*x^21 +16*x^20 +26*x^19 +12*x^18 -14*x^17 -5*x^16 +x^15 +28*x^14 -11*x^12 +x^11 +6*x^10 -4*x^9 +x^8 -3*x^7 +7*x^5 +3*x^4 +x^2 -1) / (32*x^44 +16*x^42 +64*x^40 +116*x^39 +40*x^38 -94*x^37 -122*x^36 +116*x^35 +371*x^34 +36*x^33 -258*x^32 +56*x^31 -61*x^30 +444*x^29 +91*x^28 -346*x^27 -267*x^26 +155*x^25 -55*x^24 +92*x^23 -181*x^22 -254*x^21 +188*x^20 +56*x^19 +8*x^18 -94*x^17 -37*x^16 +37*x^15 +58*x^14 -36*x^13 -37*x^12 -13*x^11 +48*x^10 +8*x^9 +5*x^8 -5*x^7 +6*x^6 +9*x^5 +5*x^4 +x^2 -1).

%Y Cf. A174249, A361250, A372221.

%K nonn,easy

%O 0,5

%A _Alois P. Heinz_, Nov 14 2023