A202536
Number of tilings of a 4 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 0, 1, 3, 3, 8, 21, 31, 70, 165, 286, 615, 1351, 2548, 5353, 11343, 22320, 46349, 96516, 193944, 400313, 826747, 1678540, 3453642, 7105102, 14498569, 29781633, 61158957, 125108639, 256763850, 526846289, 1079030715, 2213527089, 4540131569, 9304062828
Offset: 0
a(3) = 3, because there are 3 tilings of a 4 X 3 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
._____. ._____. ._._._.
| | | | |_____| |_____|
| | | | | | | | |_____|
|_|_|_| | | | | |_____|
|_____| |_|_|_| |_____|
a(4) = 3, because there are 3 tilings of a 4 X 4 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
._._____. ._____._. ._._._._.
| |_____| |_____| | | . | . |
| | . | | | | . | | |___|___|
|_|___| | | |___|_| | . | . |
|_____|_| |_|_____| |___|___|
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1, 1, 11, -9, -8, -48, 29, 23, 115, -44, -31, -173, 35, 19, 174, -18, 2, -119, 10, -14, 56, -8, 12, -19, 4, -5, 5, -1, 1, -1).
-
gf:= -(x^3+x-1) *(x^18 -3*x^15 +x^14 +7*x^12 -3*x^11 -11*x^9 +3*x^8 +12*x^6 -x^5 -6*x^3+1) *(x-1)^2 *(x^2+x+1)^2 / (x^30 -x^29 +x^28 -5*x^27 +5*x^26 -4*x^25 +19*x^24 -12*x^23 +8*x^22 -56*x^21 +14*x^20 -10*x^19 +119*x^18 -2*x^17 +18*x^16 -174*x^15 -19*x^14 -35*x^13 +173*x^12 +31*x^11 +44*x^10 -115*x^9 -23*x^8 -29*x^7 +48*x^6 +8*x^5 +9*x^4 -11*x^3 -x^2 -x+1):
a:= n-> coeff(series(gf, x, n+1),x,n);
seq(a(n), n=0..50);
A219968
Number of tilings of a 3 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 1, 1, 2, 3, 4, 8, 13, 19, 35, 58, 89, 154, 256, 405, 681, 1131, 1822, 3025, 5012, 8156, 13465, 22257, 36415, 59976, 98961, 162370, 267184, 440335, 723521, 1190237, 1960146, 3223045, 5301876, 8727650, 14355677, 23615683, 38865307, 63937660, 105184761
Offset: 0
a(6) = 8, because there are 8 tilings of a 3 X 6 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
._._._._._._. ._____._._._. ._._____._._. ._._._____._.
| | | | | | | |_____| | | | | |_____| | | | | |_____| |
| | | | | | | |_____| | | | | |_____| | | | | |_____| |
|_|_|_|_|_|_| |_____|_|_|_| |_|_____|_|_| |_|_|_____|_|
._._._._____. ._____._____. .___.___.___. ._____._____.
| | | |_____| |_____|_____| | | | | |_____|_____|
| | | |_____| |_____|_____| |___|_._|___| | | | |
|_|_|_|_____| |_____|_____| |_____|_____| |___|___|___|
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,0,3,-2,0,-1,1,0,1).
-
gf:= -(x-1)^2*(x^2+x+1)^2 / (x^9+x^7-x^6-2*x^4+3*x^3+x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);
A219969
Number of tilings of a 5 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 0, 2, 4, 8, 28, 65, 170, 456, 1177, 3068, 8016, 21028, 54696, 142908, 374527, 976022, 2551162, 6674999, 17422250, 45536000, 119053392, 310969076, 812695944, 2124175469, 5550025712, 14503987368, 37905500955, 99050326532, 258846292750, 676453686574
Offset: 0
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:
._____._. ._._____. ._._._._. ._._.___.
|_____| | | |_____| | | | | | | | | |
|_____| | | |_____| | | | | | | | |___|
|_____|_| |_|_____| |_|_|_|_| |_|_| | |
| | | | | | | | | | | | |
|___|___| |___|___| |___|___| |___|_|_|
.___.___. .___.___. .___.___. .___._._.
| | | | | | | | | | | | |
|___|___| |___|___| |___|___| |___| | |
|_____| | | |_____| | | | | | | | |_|_|
|_____| | | |_____| | | | | | | | | |
|_____|_| |_|_____| |_|_|_|_| |_|_|___|
-
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
-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 /
(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
+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):
a:= n-> coeff (series (gf, x, n+1), x, n):
seq (a(n), n=0..50);
A219970
Number of tilings of a 6 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 1, 2, 8, 21, 65, 267, 804, 2530, 9407, 29876, 96530, 337607, 1100968, 3612017, 12284181, 40538721, 133971666, 450045654, 1492504271, 4949649998, 16537407509, 54951656907, 182545459254, 608458757050, 2023395434235, 6727009100372, 22399013790607
Offset: 0
a(2) = 2, because there are 2 tilings of a 6 X 2 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
._._. .___.
| | | | |
| | | |___|
|_|_| | |
| | | |___|
| | | | |
|_|_| |___|
A219971
Number of tilings of a 7 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 0, 3, 13, 31, 170, 804, 2744, 12343, 55657, 214485, 923990, 4008013, 16241904, 68963484, 293830787, 1216393406, 5135225641, 21711233759, 90692128053, 381960417890, 1609647053073, 6749229887672, 28396742081001, 119506154362773, 501879008939056
Offset: 0
a(2) = 3, because there are 3 tilings of a 7 X 2 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
._._. .___. .___.
| | | | | | |
| | | |___| |___|
|_|_| | | | | |
| | | | | |___|
|___| |_|_| | | |
| | | | | | |
|___| |___| |_|_|
A219972
Number of tilings of an 8 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 0, 4, 19, 70, 456, 2530, 12343, 66653, 372429, 1910822, 10162059, 55583124, 291195073, 1550518247, 8363883923, 44269140576, 235940940726, 1264437357202, 6723179849519, 35846240541026, 191571690239220, 1020536749613565, 5442225134864810, 29051213879574079
Offset: 0
a(2) = 4, because there are 4 tilings of an 8 X 2 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
.___. .___. .___. .___.
| | | | | | | | | |
|___| |___| | | | | | |
| | | | | |_|_| |_|_|
|___| | | | | | | | |
| | |_|_| |___| | | |
|___| | | | | | | |_|_|
| | | | | | | | | |
|___| |_|_| |_|_| |___|
A219973
Number of tilings of a 9 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 1, 5, 35, 165, 1177, 9407, 55657, 372429, 2766100, 17671945, 118674685, 837152398, 5531816082, 37328192350, 257252634864, 1725117502706, 11666552132346, 79612474584518, 537157628862720, 3636277021289585, 24711657036469040, 167150255133278869
Offset: 0
a(2) = 5, because there are 5 tilings of a 9 X 2 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
.___. .___. .___. .___. .___.
| | | | | | | | | | | |
| | | | | | |___| |___| |___|
|_|_| |_|_| | | | | | | |
| | | | | | | | |___| |___|
| | | |___| |_|_| | | | | |
|_|_| | | | | | | | |___|
| | | |___| |___| |_|_| | | |
| | | | | | | | | | | |
|_|_| |___| |___| |___| |_|_|
A219975
Number of tilings of an n X n square using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 0, 1, 2, 3, 28, 267, 2744, 66653, 2766100, 141365332, 13305552648, 2149055591278, 493880634209398, 192321197859269019, 124351154502319720238, 122893248485909264026734, 199405053536180281080458422, 527809383857797224536981601752
Offset: 0
a(4) = 3, because there are 3 tilings of a 4 X 4 square using straight (3 X 1) trominoes and 2 X 2 tiles:
._._____. ._____._. ._._._._.
| |_____| |_____| | | . | . |
| | . | | | | . | | |___|___|
|_|___| | | |___|_| | . | . |
|_____|_| |_|_____| |___|___| .
A219974
Number of tilings of a 10 X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles.
Original entry on oeis.org
1, 0, 7, 58, 286, 3068, 29876, 214485, 1910822, 17671945, 141365332, 1225043571, 10871708273, 90688536786, 781019641877, 6800784990468, 57716277595775, 496096480109463, 4284929501626939, 36621478929328140, 314572431365390797, 2707867101849205129
Offset: 0
a(2) = 7, because there are 7 tilings of a 10 X 2 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
.___. .___. .___. .___. .___. .___. .___.
| | | | | | | | | | | | | | | | |
|___| |___| |___| | | | |___| | | | | | |
| | | | | | | |_|_| | | | |_|_| |_|_|
|___| |___| | | | | | | | | | | | | |
| | | | | |_|_| |___| |_|_| |___| | | |
|___| | | | | | | | | | | | | | |_|_|
| | |_|_| |___| |___| | | | | | | | |
|___| | | | | | | | | | |_|_| |_|_| |___|
| | | | | | | | | | | | | | | | |
|___| |_|_| |_|_| |_|_| |___| |___| |___|
Showing 1-9 of 9 results.