A234312
Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, X.
Original entry on oeis.org
1, 0, 2, 0, 4, 2, 8, 8, 16, 24, 36, 64, 88, 160, 224, 392, 576, 960, 1472, 2368, 3728, 5888, 9376, 14720, 23488, 36896, 58752, 92544, 146944, 232064, 367680, 581632, 920448, 1457152, 2305024, 3649664, 5773312, 9140224, 14460928, 22890496, 36221184, 57327616
Offset: 0
a(4) = 4:
._______. ._______. ._______. ._______.
|_. |_. | | ._| ._| |_. | ._| | ._|_. |
| | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | |
| |_| |_| |_| |_| | | |_|_| | |_| | |_|
|___|___| |___|___| |___|___| |___|___|.
a(5) = 2:
._________. ._________.
| | ._____| |_____. | |
| |_| |_. | | ._| |_| |
| |_. ._| | | |_. ._| |
|___|_| | | | | |_|___|
|_______|_| |_|_______|.
-
a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,
<0|0|0|0|1>, <2|0|0|2|0>>^n)[5, 5]:
seq(a(n), n=0..50);
-
LinearRecurrence[{0, 2, 0, 0, 2}, {1, 0, 2, 0, 4}, 50] (* Jean-François Alcover, May 28 2019 *)
A247268
Number of tilings of a 5 X n rectangle using n pentominoes of shapes Y, U, X.
Original entry on oeis.org
1, 0, 0, 1, 0, 2, 1, 0, 4, 5, 38, 22, 13, 90, 144, 457, 408, 386, 1267, 2230, 5912, 6481, 7098, 18896, 35433, 79634, 101232, 127501, 288304, 546652, 1113907, 1560356, 2148298, 4408181, 8335234, 15954116, 23827541, 35011426, 67591204, 126376945, 232719926
Offset: 0
a(3) = 1, a(5) = 2:
._____. ._________. ._________.
| ._. | |_. .___| | | |___. ._|
|_| |_| | |_| |_. | | ._| |_| |
|_. ._| , | |_. ._| | | |_. ._| |
| |_| | | ._|_| |_| |_| |_|_. |
|_____| |_|_______| |_______|_| .
-
gf:= -(x^40 +12*x^39 +36*x^38 -5*x^36 -2*x^35 +12*x^34 +54*x^33 +4*x^32 -21*x^31 -23*x^30 +4*x^29 +20*x^28 +4*x^27 -4*x^25 -7*x^24 -6*x^23 -3*x^22 +33*x^21 -7*x^20 -10*x^19 -12*x^18 -9*x^17 +12*x^16 +16*x^15 +3*x^14 -2*x^13 -2*x^12 -2*x^11 -3*x^10 +5*x^9 -2*x^6 -7*x^5 -x^4 +1) /
(x^43 +12*x^42 +36*x^41 -3*x^40 -29*x^39 -58*x^38 +12*x^37 +67*x^36 +4*x^35 -123*x^34 -99*x^33 +8*x^32 +23*x^31 -145*x^30 -52*x^29 -52*x^28 -35*x^27 -112*x^26 -99*x^25 -28*x^24 -7*x^23 -15*x^22 -99*x^21 -42*x^20 +22*x^19 +36*x^18 +26*x^17 -4*x^16 +6*x^15 +31*x^14 +5*x^13 +11*x^12 +14*x^11 +23*x^10 -5*x^9 -7*x^8 -x^7 +2*x^6 +9*x^5 +x^4 +x^3 -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..60);
A247124
Number of tilings of a 5 X n rectangle using n pentominoes of shapes I, U, X.
Original entry on oeis.org
1, 1, 1, 2, 3, 8, 14, 21, 37, 63, 122, 221, 374, 656, 1147, 2066, 3699, 6477, 11407, 20099, 35656, 63323, 111775, 197352, 348556, 616560, 1091570, 1929721, 3410509, 6028021, 10658114, 18851012, 33331681, 58927069, 104177155, 184188343, 325686763, 575858676
Offset: 0
a(4) = 3:
._______. ._______. ._______.
| | | | | | | ._. | | ._. | |
| | | | | | |_| |_| |_| |_| |
| | | | | | |_. ._| |_. ._| |
| | | | | | | |_| | | |_| | |
|_|_|_|_| |_|_____| |_____|_| .
-
gf:= -(x-1)^2 *(x^4+x^3+x^2+x+1)^2 /
(x^15 +x^13 +x^11 -3*x^10 -2*x^8 -2*x^6 +6*x^5 +x^3 +x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);
A278964
Number of tilings of a 6 X n rectangle using pentominoes of shapes L, U, X and monominoes.
Original entry on oeis.org
1, 1, 31, 432, 9851, 150749, 2392788, 38885816, 646657365, 10599739324, 173226495187, 2835753730682, 46501906844608, 762113847626540, 12484929995911672, 204541850515528677, 3351522453919340317, 54915791725126323755, 899777273419073185968
Offset: 0
a(2) = 31:
.___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
|_|_| | ._| | ._| |_|_| |_|_| |_|_| |_. | |_. | |_|_| |_|_| |_|_|
|_|_| | |_| | |_| | ._| |_|_| |_|_| |_| | |_| | |_. | |_|_| |_|_|
|_|_| |___| |___| | |_| | ._| |_|_| |___| |___| |_| | |_. | |_|_|
|_|_| | ._| |_|_| |___| | |_| | ._| |_. | |_|_| |___| |_| | |_. |
|_|_| | |_| |_|_| |_|_| |___| | |_| |_| | |_|_| |_|_| |___| |_| |
|_|_| |___| |_|_| |_|_| |_|_| |___| |___| |_|_| |_|_| |_|_| |___|
.___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
| ._| |_. | | |_| |_|_| |_|_| |_. | |_. | |_|_| |_. | |_|_| |_|_|
| |_| |_| | | |_| | |_| |_|_| | | | |_| | |_. | |_| | |_. | |_|_|
|___| |___| | |_| | |_| | |_| | | | | | | | | | |_| | |_| | |_. |
|_. | | ._| |___| | |_| | |_| | |_| | |_| | | | |_|_| |_| | |_| |
|_| | | |_| |_|_| |___| | |_| |___| | |_| | |_| |_|_| |_|_| |_| |
|___| |___| |_|_| |_|_| |___| |_|_| |___| |___| |_|_| |_|_| |_|_|
.___. .___. .___. .___. .___. .___. .___. .___. .___.
| ._| |_|_| |_|_| | ._| | ._| |_|_| |_|_| |_|_| |_| |
| |_| | ._| |_|_| | | | | |_| | ._| |_|_| |_| | |_| |
| |_| | |_| | ._| | | | | | | | | | |_| | |_| | |_| |
|_|_| | |_| | |_| |_| | |_| | | | | |_| | |_| | |___|
|_|_| |_|_| | |_| |___| |_| | |_| | |_| | |___| |_|_|
|_|_| |_|_| |_|_| |_|_| |___| |___| |___| |_|_| |_|_| .
A286347
Number of tilings of a 6 X n rectangle using pentominoes of shapes X, Y, T and monominoes.
Original entry on oeis.org
1, 1, 15, 188, 3763, 54368, 790504, 11771179, 176285795, 2627027061, 39172906410, 584774465361, 8725442013820, 130117410190768, 1940656499251689, 28950613608637003, 431889355175650435, 6442566792491842258, 96102610815876310611, 1433558139553752297236
Offset: 0
A286391
Number of tilings of a 6 X n rectangle using pentominoes of shapes X, Y, Z, T and monominoes.
Original entry on oeis.org
1, 1, 15, 344, 6835, 119392, 2136080, 38329669, 688355077, 12374676591, 222673114420, 4005005917279, 72004800735036, 1294444641691072, 23273376392535695, 418470613768822683, 7524342117053834729, 135288703461639048592, 2432489164334993344213
Offset: 0
a(2) = 15:
.___. .___. .___. .___. .___. .___. .___. .___. .___.
|_|_| | |_| |_| | | |_| |_|_| |_|_| |_|_| |_|_| | |_|
|_|_| | ._| |_. | | ._| | |_| |_|_| |_|_| | |_| | |_|
|_|_| | | | | | | | |_| | ._| | |_| | |_| | |_| | ._|
|_|_| |_| | | |_| |_|_| | |_| | ._| | |_| | ._| |_|_|
|_|_| |_. | | ._| |_|_| |_|_| | |_| | ._| |_|_| |_|_|
|_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_| |_|_|
.___. .___. .___. .___. .___. .___.
|_| | |_|_| |_|_| |_|_| |_|_| |_| |
|_. | |_| | |_|_| |_|_| |_| | |_| |
|_| | |_. | |_| | |_| | |_| | |_. |
|_|_| |_| | |_. | |_| | |_. | |_|_|
|_|_| |_|_| |_| | |_. | |_|_| |_|_|
|_|_| |_|_| |_|_| |_|_| |_|_| |_|_| .
Showing 1-6 of 6 results.