A219862
Number of tilings of a 4 X n rectangle using dominoes and straight (3 X 1) trominoes.
Original entry on oeis.org
1, 1, 7, 41, 184, 1069, 5624, 29907, 161800, 862953, 4631107, 24832532, 133028028, 713283085, 3822965706, 20491221900, 109840081931, 588746006676, 3155783700063, 16915482096570, 90669231898345, 486001022349368, 2605035346917456, 13963368769216664
Offset: 0
a(2) = 7, because there are 7 tilings of a 4 X 2 rectangle using dominoes and straight (3 X 1) trominoes:
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- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, 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).
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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) /
(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):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);
A219868
Number of tilings of a 5 X n rectangle using dominoes and straight (3 X 1) trominoes.
Original entry on oeis.org
1, 2, 15, 143, 1069, 9612, 82634, 707903, 6155230, 53074167, 458901955, 3968548571, 34293794622, 296497719200, 2563023778341, 22155553744380, 191528023181443, 1655649630655481, 14312296720331975, 123722938782920490, 1069522524403416934, 9245501816227527991
Offset: 0
a(2) = 15, because there are 15 tilings of a 5 X 2 rectangle using dominoes and straight (3 X 1) trominoes:
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.___. .___. .___. .___. ._._. ._._. .___.
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A219869
Number of tilings of a 6 X n rectangle using dominoes and straight (3 X 1) trominoes.
Original entry on oeis.org
1, 2, 30, 472, 5624, 82634, 1143834, 15859323, 223026907, 3105936858, 43427618185, 606962113407, 8477372675760, 118475405895601, 1655302050975409, 23128348583334118, 323168394713538001, 4515400902297318144, 63091618970759626487, 881547198837947075202
Offset: 0
a(1) = 2, because there are 2 tilings of a 6 X 1 rectangle using dominoes and straight (3 X 1) trominoes:
._. ._.
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A219870
Number of tilings of a 7 X n rectangle using dominoes and straight (3 X 1) trominoes.
Original entry on oeis.org
1, 3, 60, 1562, 29907, 707903, 15859323, 354859954, 8061851335, 181382499259, 4095897476480, 92476840837163, 2086314577400136, 47096964973772265, 1062921614745008697, 23989328157229264043, 541446343762904191567, 12220135872229640539724
Offset: 0
a(1) = 3, because there are 3 tilings of a 7 X 1 rectangle using dominoes and straight (3 X 1) trominoes:
._. ._. ._.
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A219871
Number of tilings of an 8 X n rectangle using dominoes and straight (3 X 1) trominoes.
Original entry on oeis.org
1, 4, 123, 5233, 161800, 6155230, 223026907, 8061851335, 295743829064, 10750194262698, 392127322268846, 14301080443962397, 521186161714401053, 19005127884701974465, 692863015545590519355, 25260000472133325186613, 920954865421382858310101
Offset: 0
a(1) = 4, because there are 4 tilings of an 8 X 1 rectangle using dominoes and straight (3 X 1) trominoes:
._. ._. ._. ._.
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A219872
Number of tilings of a 9 X n rectangle using dominoes and straight (3 X 1) trominoes.
Original entry on oeis.org
1, 5, 249, 17395, 862953, 53074167, 3105936858, 181382499259, 10750194262698, 631206895803116, 37197513508819191, 2191568660367709311, 129026759279686110418, 7600920859497795763717, 447657086066162084823497, 26365510156948695071183306
Offset: 0
a(1) = 5, because there are 5 tilings of a 9 X 1 rectangle using dominoes and straight (3 X 1) trominoes:
._. ._. ._. ._. ._.
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A219873
Number of tilings of a 10 X n rectangle using dominoes and straight (3 X 1) trominoes.
Original entry on oeis.org
1, 7, 506, 58002, 4631107, 458901955, 43427618185, 4095897476480, 392127322268846, 37197513508819191, 3541054185616706122, 337034306456285944790, 32054951461147041813900, 3050527403495695959369072, 290235791993563362905189432, 27614468898770593799634611889
Offset: 0
a(1) = 7, because there are 7 tilings of a 10 X 1 rectangle using dominoes and straight (3 X 1) trominoes:
._. ._. ._. ._. ._. ._. ._.
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A278815
Number of tilings of a 2 X n grid with monomers, dimers, and trimers.
Original entry on oeis.org
1, 2, 7, 29, 109, 416, 1596, 6105, 23362, 89415, 342193, 1309593, 5011920, 19180976, 73406985, 280933906, 1075154535, 4114694797, 15747237101, 60265824784, 230641706484, 882682631025, 3378090801226, 12928199853783, 49477163668857, 189352713633433
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Kathryn Haymaker and Sara Robertson, Counting Colorful Tilings of Rectangular Arrays, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.8, Corollary 2.
- Index entries for linear recurrences with constant coefficients, signature (3,2,5,-2,0,-1).
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a:=[1,2,7,29,109,416];; for n in [7..30] do a[n]:=3*a[n-1]+2*a[n-2] +5*a[n-3]-2*a[n-4]-a[n-6]; od; a; # G. C. Greubel, Oct 28 2019
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R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+2*x^4+x^6) )); // G. C. Greubel, Oct 28 2019
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seq(coeff(series((1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+2*x^4+x^6), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 28 2019
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LinearRecurrence[{3,2,5,-2,0,-1}, {1,2,7,29,109,416}, 30] (* G. C. Greubel, Oct 28 2019 *)
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my(x='x+O('x^30)); Vec((1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+ 2*x^4 +x^6)) \\ G. C. Greubel, Oct 28 2019
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def A278815_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( (1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+2*x^4+x^6) ).list()
A278815_list(30) # G. C. Greubel, Oct 28 2019
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