A234312 -id:A234312 - 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.

A247680 Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L, F.

Original entry on oeis.org

1, 1, 3, 5, 21, 82, 249, 688, 1879, 5690, 17932, 55271, 164427, 485348, 1451110, 4395114, 13313135, 40073992, 120200822, 360897368, 1086543152, 3274191643, 9858847241, 29657925485, 89206237151, 268435863317, 808022052324, 2432169981689, 7319562671432

Offset: 0

Alois P. Heinz, Sep 22 2014

nonn

			a(3) = 5:
._____.  ._____.  ._____.  ._____.  ._____.
| | | |  | |_. |  | ._| |  | | ._|  |_. | |
| | | |  | | | |  | | | |  | | | |  | | | |
| | | |  | | | |  | | | |  | | | |  | | | |
| | | |  | | |_|  |_| | |  | |_| |  | |_| |
|_|_|_|  |_|___|  |___|_|  |_|___|  |___|_|.
a(4) = 21:
._______.  ._______.
|_. |_. |  | ._| ._|
| |_. | |  | |_. | |
|_. |_| |  | | |_| |
| |___|_|  |_| ._| |
|_______|  |___|___| ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, G.f. and the recurrence (of order 324)
Wikipedia, Pentomino

Cf. A174249, A233427, A234312, A234931, A247443, A249762, A257866.

a(n) ~ c * d^n, where d = 3.009533036298033336764263169394953980849599088993157702490314631810945318907..., c = 0.29272000293879867768013500033525343337565088925220444775140709413075274... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015

A247125 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, X.

Original entry on oeis.org

1, 0, 2, 1, 16, 10, 59, 60, 330, 397, 1520, 2218, 7875, 12820, 39250, 70045, 202168, 384866, 1038051, 2073580, 5385754, 11156701, 28015232, 59580154, 146333795, 317517636, 766142242, 1686735709, 4019319048, 8946988370, 21116854115, 47386013020, 111065223914

Offset: 0

Alois P. Heinz, Nov 19 2014

nonn

			a(4) = 16:
._______.     ._______.     ._______.
| ._____|     | ._____|     | ._| ._|
|_| |_. |     |_| |_. |     | | | | |
|_. ._| |     |_. ._| |     | | | | |
| |_|___|     | |_| | |     |_| |_| |
|_______| (2) |_____|_| (4) |___|___| (4)
._______.     ._______.
| ._____|     | ._____|
|_| ._. |     |_|_. | |
| |_| |_|     | ._| | |
|_____| |     | |___| |
|_______| (2) |___|___| (4) .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,2,1,12,6,2).

Cf. A174249, A233427, A234312, A247124, A247268.

a:= n-> (<<0|1|0|0|0|0>, <0|0|1|0|0|0>, <0|0|0|1|0|0>,
          <0|0|0|0|1|0>, <0|0|0|0|0|1>, <2|6|12|1|2|0>>^n)[6,6]:
seq(a(n), n=0..40);

G.f.: -1/(2*x^6+6*x^5+12*x^4+x^3+2*x^2-1).

A264812 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, X.

Original entry on oeis.org

1, 1, 3, 5, 13, 52, 123, 366, 909, 2444, 7108, 19157, 53957, 146826, 400704, 1115852, 3059907, 8475420, 23369304, 64225984, 177572352, 488839323, 1349102071, 3722419367, 10255126169, 28303059509, 78013005366, 215160477217, 593488173404, 1636220978049

Offset: 0

Alois P. Heinz, Nov 25 2015

nonn

			a(4) = 13:
._______.      ._______.      ._______.      ._______.
| | | | |      |   |   |      |   | | |      |   ._| |
| | | | |      | ._| ._|      | ._| | |      |___|   |
| | | | |      |_| |_| |      |_| | | |      |   |___|
| | | | | (1)  |   |   | (4)  |   | | | (6)  | ._|   | (2)
|_|_|_|_|      |___|___|      |_ _|_|_|      |_|_____|    .
a(5) = 52:
._________.
|   |_.   |
| ._| |___|
|_|_   _| |
|   |_|   | (2)  ...
|_____|___|          .

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

Cf. A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A278330.

A278330 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, U, X.

Original entry on oeis.org

1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984, 7019, 11148, 35686, 62181, 182776, 339350, 942507, 1841208, 4887096, 9921685, 25442304, 53190380, 132928715, 284198328, 696276202, 1514363221, 3654567764, 8053235650, 19212546163, 42762014028, 101125071372

Offset: 0

Alois P. Heinz, Nov 18 2016

			a(2) = 2,          a(3) = 1:
.___.   .___.      ._____.
|   |   |   |      | ._. |
| ._|   |_. |      |_| |_|
|_| |   | |_|      |_   _|
|   |   |   |      | |_| |
|___|   |___|      |_____| .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,2,2,8,4,21,-8,-4,-6,0,-16,-8).

Cf. A079978, A174249, A233427, A234312, A234931, A247124, A247268, A247443, A249762, A264765, A264812.

a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
    [-8, -16, 0, -6, -4, -8, 21, 4, 8, 2, 2, 0][j], 0)))^n.
    <<1, 0, 2, 1, 12, 10, 59, 52, 276, 349, 1404, 1984>>)[1, 1]:
seq(a(n), n=0..35);

G.f.: -(4*x^6+x^3-1) / (8*x^12 +16*x^11 +6*x^9 +4*x^8 +8*x^7 -21*x^6 -4*x^5 -8*x^4 -2*x^3 -2*x^2+1).

a(n) mod 2 = A079978(n).

A077909 Expansion of 1/((1-x)(1+x+x^2+2x^3)).

Original entry on oeis.org

1, 0, 0, -1, 2, 0, 1, -4, 4, -1, 6, -12, 9, -8, 24, -33, 26, -40, 81, -92, 92, -161, 254, -276, 345, -576, 784, -897, 1266, -1936, 2465, -3060, 4468, -6337, 7990, -10588, 15273, -20664, 26568, -36449, 51210, -67896, 89585, -124108, 170316, -225377, 303278, -418532, 566009, -754032, 1025088

Offset: 0

N. J. A. Sloane, Nov 17 2002

The absolute value of a(n) is the number of tilings of a 5 X n rectangle using n pentominoes of shapes N, U, X. |a(3)| = 1, |a(4)| = 2:
  .___.     ._____.  ._____.
  | .. |     | .. | |  | | ._. |
  || ||     || || |  | || ||
  |. .|  ,  | .| .|  |. |. |
  | || |     | | || |  | |_| | |
  |___|     ||____|  |___|_|. - Alois P. Heinz, Jan 03 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, signature (0,0,-1,2)

Partial sums of A077976.

Cf. A174249, A233427, A234312, A234931, A247126.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <2|-1|0|0>>^n.
        <<1, 0, 0, -1>>)[1, 1]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 20 2013

CoefficientList[1/(1+x^3-2*x^4) + O[x]^60, x] (* Jean-François Alcover, Jun 08 2015, after Arkadiusz Wesolowski *)

Vec( 1/((1-x)*(1+x+x^2+2*x^3)) +O(x^66)) \\ Joerg Arndt, Aug 28 2013

a(n) = (-1)^n*sum(A128099(n-2*k, n-3*k), k=0..floor(n/3)). - Johannes W. Meijer, Aug 28 2013

G.f.: 1/(1 + x^3 - 2*x^4). - Arkadiusz Wesolowski, Nov 20 2013

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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Maple

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A247680 Number of tilings of a 5 X n rectangle using n pentominoes of shapes W, I, L, F.

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A247125 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, U, X.

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Maple

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A264812 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, I, X.

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A278330 Number of tilings of a 5 X n rectangle using n pentominoes of shapes P, U, X.

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Maple

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A077909 Expansion of 1/((1-x)*(1+x+x^2+2*x^3)).

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PARI

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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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A077909 Expansion of 1/((1-x)(1+x+x^2+2x^3)).