A247680 -id:A247680 - cp's OEIS frontend

A234931 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, N.

1, 0, 0, 0, 2, 0, 0, 0, 4, 0, 8, 0, 16, 0, 40, 0, 64, 16, 200, 96, 504, 464, 1528, 1664, 4376, 5616, 12792, 18192, 38264, 58384, 115832, 186368, 355808, 589344, 1095408, 1853664, 3383656, 5802016, 10470376, 18125280, 32461312, 56552736, 100782696, 176318464

Offset: 0

Alois P. Heinz, Jan 01 2014

nonn

			a(4) = 2:
._______.   ._______.
| | ._. |   | ._. | |
| |_| |_|   |_| |_| |
|_. |_. |   | ._| ._|
| |_| | |   | | |_| |
|_____|_|   |_|_____|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A247680, A247126.

G.f.: (4*x^20 +4*x^18 +8*x^16 -3*x^14 +4*x^13 -5*x^12 -2*x^11 +3*x^10 -2*x^9 +6*x^8 -2*x^7 +2*x^6 -2*x^5 -x^4 +2*x -1) / (-8*x^22 -28*x^20 -6*x^18 +8*x^17 +26*x^16 +4*x^15 +7*x^14 -8*x^13 -9*x^12 -14*x^11 +7*x^10 +2*x^9 +8*x^8 -2*x^7 +2*x^6 -6*x^5 +x^4 +2*x -1).

A247443 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, L, Y.

Original entry on oeis.org

1, 0, 2, 0, 6, 16, 32, 104, 186, 800, 1700, 4836, 11186, 29940, 84388, 208808, 563364, 1391664, 3787510, 9824684, 25712276, 66815444, 173151378, 457266220, 1188536784, 3113743272, 8087358736, 21152284376, 55283003950, 144314582896, 376852311434, 982507243820

Offset: 0

Alois P. Heinz, Sep 23 2014

nonn

			a(5) = 16:
._._______.        ._______._.
| | ._____|        |_. .___| |
| |_| ._| |        | |_| ._| |
| |_. |_. |        | |_. |_. |
|___|_| | |        | ._|_| |_|
|_______|_| (*8)   |_|_______| (*8)  .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alois P. Heinz, G.f. and Maple program for A247443
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A234931, A247680, A248102, A249762.

Maple
```
# Maple program: see link.
```

G.f.: see link.

A249762 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, I, L.

Original entry on oeis.org

1, 1, 3, 5, 13, 50, 133, 360, 875, 2254, 6336, 17331, 47199, 124476, 330344, 889454, 2400961, 6485476, 17392906, 46616158, 125153478, 336529923, 905611165, 2434088873, 6539233985, 17567977887, 47214493386, 126927551197, 341177175540, 916960655233

Offset: 0

Alois P. Heinz, Dec 03 2014

nonn

			a(4) = 13:
._______.     ._______.     ._______.
| | | | |     | ._| ._|     | | ._| |
| | | | |     | |_. | |     | | | | |
| | | | |     | | |_| |     | | | | |
| | | | |     |_| ._| |     | |_| | |
|_|_|_|_| (1) |___|___| (2) |_|___|_| (2)
._______.     ._______.     ._______.
| ._| | |     | ._| ._|     | ._|_. |
| | | | |     | | | | |     | | | | |
| | | | |     | | | | |     | | | | |
|_| | | |     |_| |_| |     |_| | |_|
|___|_|_| (4) |___|___| (2) |___|___| (2) .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, G.f. and the recurrence (of order 315)
Wikipedia, Pentomino

Cf. A174249, A233427, A247443, A247680, A251617, A251737, A257866, A264765.

a(n) ~ c * d^n, where d = 2.6877447474867836174937272605197376719631593933016281800670782370745298422..., c = 0.3236150736074998563483897952085529328299218049725560430481595704054051228... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015

A251617 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, I, L, U.

Original entry on oeis.org

1, 1, 3, 5, 19, 74, 219, 628, 1749, 5486, 17448, 53383, 160169, 479908, 1468366, 4512092, 13782535, 41855766, 127112554, 387469920, 1182800866, 3606789463, 10983721059, 33445214911, 101911804705, 310658892951, 946813182854, 2884825285301, 8789233684468

Offset: 0

Alois P. Heinz, Dec 05 2014

nonn

			a(4) = 19 = 13 + 4 + 2 = A249762(4) + 4 + 2:
._______.     ._______.
|_____. |     | ._____|
| | ._|_|     |_| ._. |
| | |_. |     | |_| |_|
| |___| |     |_____| |
|___|___| (4) |_______| (2) .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A247443, A247680, A249762, A251737.

A251737 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, I.

Original entry on oeis.org

1, 1, 3, 5, 17, 66, 181, 508, 1283, 3664, 10812, 31171, 88565, 245524, 692416, 1968532, 5609977, 15928174, 44982196, 127190716, 360208608, 1021611491, 2896270245, 8202605953, 23226285083, 65780006703, 186369631872, 528047092459, 1495905404102, 4237308534243

Offset: 0

Alois P. Heinz, Dec 07 2014

nonn

			a(4) = 17:
._______.     ._______.     ._______.     ._______.
|_____. |     | ._____|     | | | | |     | | ._| |
| | ._|_|     |_| ._. |     | | | | |     | | | | |
| | |_. |     | |_| |_|     | | | | |     | | | | |
| |___| |     |_____| |     | | | | |     | |_| | |
|___|___| (4) |_______| (2) |_|_|_|_| (1) |_|___|_| (2)
._______.     ._______.     ._______.
| ._| | |     | ._| ._|     | ._|_. |
| | | | |     | | | | |     | | | | |
| | | | |     | | | | |     | | | | |
|_| | | |     |_| |_| |     |_| | |_|
|___|_|_| (4) |___|___| (2) |___|___| (2) .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A247443, A247680, A249762, A251617.

A257866 Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L.

Original entry on oeis.org

1, 1, 3, 5, 19, 74, 209, 572, 1479, 4304, 13002, 38315, 109651, 308982, 884120, 2560952, 7428183, 21413028, 61433280, 176415916, 507985116, 1464725431, 4220293147, 12145885239, 34945690653, 100586823613, 289649303130, 834087280681, 2401368817168, 6912685066843

Offset: 0

Alois P. Heinz, May 11 2015

nonn

			a(3) = 5:
._____. ._____. ._____. ._____. ._____.
| | | | | |_. | | ._| | | | ._| |_. | |
| | | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | | | | | |
| | | | | | |_| |_| | | | |_| | | |_| |
|_|_|_| |_|___| |___|_| |_|___| |___|_|.
a(4) = 19:
._______. ._______.
|_. |_. | | | ._| |
| |_. | | | | | | |
|_. |_| | | | | | |
| |___|_| | |_| | |
|_______| |_|___|_| ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, G.f. and the recurrence (of order 315)
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A234931, A247443, A247680, A249762.

a(n) ~ c * d^n, where d = 2.878962978866730659679600165158895088546680936475540731494833253735549346144..., c = 0.33249894796240209167801000207088312509480543003269025485052861968247997... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015

A248102 Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shapes N, Y.

Original entry on oeis.org

1, 0, 0, 18, 24, 238, 842, 4360, 25900, 112178, 613140, 2941170, 14789274, 74895336, 369603312, 1866863986, 9294391952, 46543456838, 233028690018, 1164275409976, 5830080180396, 29149585256266, 145845002931724, 729627382873090, 3649578988919810

Offset: 0

Alois P. Heinz, Oct 01 2014

nonn

			a(3) = 18:
._______._._.        .___._______.        .___._______.
|_. .___| | |        |_. |___. ._|        |_. |___. ._|
| |_| | ._| |        | |_____|_| |        | |_____|_| |
| ._| | |_. |        | |___. ._| |        | |___. |_. |
| | ._|_| |_|        | ._| |_|_. |        | ._| |___| |
|_|_|_______| (*2)   |_|_______|_| (*8)   |_|_______|_| (*8)  .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A234931, A247680, A247443.

G.f.: (8*x^15 -52*x^14 -64*x^13 +1087*x^12 -2822*x^11 +2369*x^10 +810*x^9 -2047*x^8 +300*x^7 +122*x^6 +208*x^5 +x^4 +6*x^3 +9*x^2 +2*x -1) / (24*x^15 +4*x^14 -680*x^13 +2673*x^12 -4212*x^11 +2139*x^10 +1574*x^9 -2141*x^8 +456*x^7 -160*x^6 +236*x^5 -11*x^4 +24*x^3 +9*x^2 +2*x -1).

cp's OEIS Frontend

A234931 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, U, N.

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A247443 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, L, Y.

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A249762 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, I, L.

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A251617 Number of tilings of a 5 X n rectangle using n pentominoes of shapes F, I, L, U.

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A251737 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, I.

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A257866 Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L.

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A248102 Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shapes N, Y.

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