A233427 -id:A233427 - cp's OEIS frontend

A174249 Number of tilings of a 5 X n rectangle with n pentominoes of any shape.

1, 1, 5, 56, 501, 4006, 27950, 214689, 1696781, 13205354, 101698212, 782267786, 6048166230, 46799177380, 361683136647, 2793722300087, 21583392631817, 166790059833039, 1288885349447958, 9959188643348952, 76953117224941654, 594617039453764617, 4594660583890506956

Offset: 0

Bob Harris (me13013(AT)gmail.com), Mar 13 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jessica Gonzalez, Illustration for a(2) = 5
R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings
Vaclav Kotesovec, G.f. and the recurrence (of order 324)
Wikipedia, Pentomino
Index entries for linear recurrences with constant coefficients, order 324.

Cf. A134438, A174248, A174250, A174251, A174252, A174253, A246902, A278456.
Column k=5 of A233427.
Row sums of A247702, A247703, A247704, A247705, A247706, A247707, A247708, A247709, A247710, A247711, A247712, A247713.

a(n) ~ c * d^n, where d =
7.727036840800092392128639105511391434436212757335030092041375597587338371937..., c =
0.13364973920881772493778581621701653927538155984099992758656160782495174... (1/d is the root of the denominator, see g.f.). - Vaclav Kotesovec, May 19 2015

a(0) prepended, a(11)-a(22) from Alois P. Heinz, Dec 05 2013

A099390 Array T(m,n) read by antidiagonals: number of domino tilings (or dimer tilings) of the m X n grid (or m X n rectangle), for m>=1, n>=1.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 1, 3, 3, 1, 0, 5, 0, 5, 0, 1, 8, 11, 11, 8, 1, 0, 13, 0, 36, 0, 13, 0, 1, 21, 41, 95, 95, 41, 21, 1, 0, 34, 0, 281, 0, 281, 0, 34, 0, 1, 55, 153, 781, 1183, 1183, 781, 153, 55, 1, 0, 89, 0, 2245, 0, 6728, 0, 2245, 0, 89, 0, 1, 144, 571, 6336, 14824, 31529, 31529, 14824, 6336, 571, 144, 1

Offset: 1

Ralf Stephan, Oct 16 2004

There are many versions of this array (or triangle) in the OEIS. This is the main entry, which ideally collects together all the references to the literature and to other versions in the OEIS. But see A004003 for further information. - N. J. A. Sloane, Mar 14 2015

			0,  1,  0,   1,    0,    1, ...
1,  2,  3,   5,    8,   13, ...
0,  3,  0,  11,    0,   41, ...
1,  5, 11,  36,   95,  281, ...
0,  8,  0,  95,    0, 1183, ...
1, 13, 41, 281, 1183, 6728, ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 406-412.
P. E. John, H. Sachs, and H. Zernitz, Problem 5. Domino covers in square chessboards, Zastosowania Matematyki (Applicationes Mathematicae) XIX 3-4 (1987), 635-641.
R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 2nd ed., pp. 547 and 570.
Darko Veljan, Kombinatorika: s teorijom grafova (Croatian) (Combinatorics with Graph Theory) mentions the value 12988816 = 2^4*901^2 for the 8 X 8 case on page 4.

Alois P. Heinz, Table of n, a(n) for n = 1..1035
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles, arXiv:math/0610925 [math.CO], 2006.
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, Journal of Combinatorial Theory, Series A, Volume 77, Issue 1, January 1997, Pages 67-97.
Henry Cohn, 2-adic behavior of numbers of domino tilings, arXiv:math/0008222 [math.CO], 2000.
Henry Cohn, 2-adic behavior of numbers of domino tilings, Electronic Journal of Combinatorics, 6 (1999), #R14.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Steven R. Finch, The Dimer Problem [From Steven Finch, Apr 20 2019]
Steven R. Finch, Two Dimensional Monomer Dimer Constant [Broken link]
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 363.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
Peter E. John and Horst Sachs, On a strange observation in the theory of the dimer problem, arXiv:math/9801094 [math.CO], 1998.
Peter E. John and Horst Sachs, On a strange observation in the theory of the dimer problem, Discrete Math. 216 (2000), no. 1-3, 211-219.
Yuhi Kamio, Junnosuke Koizumi, and Toshihiko Nakazawa, Quadratic residues and domino tilings, arXiv:2311.13597 [math.NT], 2023.
David Klarner and Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52, Table 1.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
P. W. Kasteleyn, The statistics of dimers on a lattice, I. the number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225.
Douglas M. McKenna, The Art of Space-Filling Domino Curves, Bridges Conference Proceedings, 2024, pp. 319-326.
L. Pachter, Combinatorial approaches and conjectures for 2-divisibility problems concerning domino tilings of polyominoes, Electronic Journal of Combinatorics 4 (1997), #R29.
J. Propp, Dimers and Dominoes
J. Propp, Enumeration of Matchings: Problems and Progress, arXiv:math/9904150v2 [math.CO], 1999.
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
R. C. Read, A Note on Tiling Rectangles with Dominoes, The Fibonacci Quarterly, 18.1 (1980), 24-27.
R. P. Stanley, A combinatorial miscellany
H. N. V. Temperley and Michael E. Fisher, Dimer problem in statistical mechanics -- an exact result, Philos. Mag. (8) 6 (1961), 1061-1063.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Domino Tiling
Eric Weisstein, Illustration for T(4,4) = 36, from Domino Tilings web page (see previous link) [Included with permission]
Index entries for sequences related to dominoes

See A187596 for another version (with m >= 0, n >= 0). See A187616 for a triangular version. See also A187617, A187618.
See also A004003 for more literature on the dimer problem.
Rows 2-13, 16 (without zeros) are A000045, A001835, A005178, A003775, A028468, A028469, A028470, A028471, A028472, A028473, A028474, A241908, A340532.
Main diagonal is A004003.
Cf. A103997, A103999, A233320, A230031, A233427.

(Maple code for the even-numbered rows from N. J. A. Sloane, Mar 15 2015. This is not totally satisfactory since it uses floating point. However, it is useful for getting the initial values quickly.)
Digits:=100;
p:=evalf(Pi);
z:=proc(h,d) global p; evalf(cos( h*p/(2*d+1) )); end;
T:=proc(m,n) global z; round(mul( mul( 4*z(h,m)^2+4*z(k,n)^2, k=1..n), h=1..m)); end;
[seq(T(1,n),n=0..10)]; # A001519
[seq(T(2,n),n=0..10)]; # A188899
[seq(T(3,n),n=0..10)]; # A256044
[seq(T(n,n),n=0..10)]; # A004003

T[?OddQ, ?OddQ] = 0;
T[m_, n_] := Product[2*(2+Cos[2j*Pi/(m+1)]+Cos[2k*Pi/(n+1)]), {k, 1, n/2}, {j, 1, m/2}];
Flatten[Table[Round[T[m-n+1, n]], {m, 1, 12}, {n, 1, m}]] (* Jean-François Alcover, Nov 25 2011, updated May 28 2022 *)

{T(n, k) = sqrtint(abs(polresultant(polchebyshev(n, 2, x/2), polchebyshev(k, 2, I*x/2))))} \\ Seiichi Manyama, Apr 13 2020

T(m, n) = Product_{j=1..ceiling(m/2)} Product_{k=1..ceiling(n/2)} (4*cos(j*Pi/(m+1))^2 + 4*cos(k*Pi/(n+1))^2).

Old link fixed and new link added by Frans J. Faase, Feb 04 2009

Entry edited by N. J. A. Sloane, Mar 15 2015

A230031 Number A(n,k) of tilings of a k X n rectangle using tetrominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 4, 0, 4, 0, 1, 1, 0, 0, 23, 23, 0, 0, 1, 1, 0, 9, 0, 117, 0, 9, 0, 1, 1, 1, 0, 0, 454, 454, 0, 0, 1, 1, 1, 0, 25, 0, 2003, 0, 2003, 0, 25, 0, 1, 1, 0, 0, 997, 9157, 0, 0, 9157, 997, 0, 0, 1

Offset: 0

Alois P. Heinz, Nov 29 2013

			A(4,2) = A(2,4) = 4:
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |_______|  | |___. |  | .___| |
  |___|___|  |_______|  |_____|_|  |_|_____|.
Square array A(n,k) begins:
  1, 1,  1,   1,     1,      1,        1,         1,           1, ...
  1, 0,  0,   0,     1,      0,        0,         0,           1, ...
  1, 0,  1,   0,     4,      0,        9,         0,          25, ...
  1, 0,  0,   0,    23,      0,        0,         0,         997, ...
  1, 1,  4,  23,   117,    454,     2003,      9157,       40899, ...
  1, 0,  0,   0,   454,      0,        0,         0,      800290, ...
  1, 0,  9,   0,  2003,      0,   178939,         0,    22483347, ...
  1, 0,  0,   0,  9157,      0,        0,         0,   657253434, ...
  1, 1, 25, 997, 40899, 800290, 22483347, 657253434, 19077209438, ...

Liang Kai, Antidiagonals n = 0..27, flattened (Antidiagonals n = 0..20 from Alois P. Heinz)
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
R. S. Harris, Counting Polyomino Tilings
Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
Wikipedia, Tetris
Wikipedia, Tetromino

Columns (or rows) include: A000012, A007598, A232757, A174248, A232758, A232684, A232759, A232698, A247113, A232722.
Bisection of main diagonal (even part) gives A263425.
Cf. A099390, A233320, A233427.

A(n,k) = 0 <=> n*k mod 4 > 0.

A233320 Number A(n,k) of tilings of a k X n rectangle using trominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 3, 3, 0, 1, 1, 0, 0, 10, 0, 0, 1, 1, 1, 0, 23, 23, 0, 1, 1, 1, 0, 11, 62, 0, 62, 11, 0, 1, 1, 0, 0, 170, 0, 0, 170, 0, 0, 1, 1, 1, 0, 441, 939, 0, 939, 441, 0, 1, 1, 1, 0, 41, 1173, 0, 8342, 8342, 0, 1173, 41, 0, 1

Offset: 0

Alois P. Heinz, Dec 07 2013

Every row and column satisfies a linear recurrence. - Peter Kagey, Jul 17 2019

			Square array A(n,k) begins:
  1, 1,  1,    1,   1,    1,       1, ...
  1, 0,  0,    1,   0,    0,       1, ...
  1, 0,  0,    3,   0,    0,      11, ...
  1, 1,  3,   10,  23,   62,     170, ...
  1, 0,  0,   23,   0,    0,     939, ...
  1, 0,  0,   62,   0,    0,    8342, ...
  1, 1, 11,  170, 939, 8342,   80092, ...
  1, 0,  0,  441,   0,    0,  614581, ...
  1, 0,  0, 1173,   0,    0, 5271923, ...

Liang Kai, Antidiagonals n = 0..35, flattened (Antidiagonals 0..28 from Alois P. Heinz)
Wikipedia, Tromino

Columns (or rows) include: A000012, A001835, A134438, A233339, A233340, A233290, A233343, A269958, A215826, A269959, A269664.

Cf. A099390, A230031, A233427, A270061, A364457.

A(n,k) = 0 <=> n*k mod 3 > 0.

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

Original entry on oeis.org

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).

A278657 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape and monominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 2, 25, 50, 25, 2, 1, 1, 3, 50, 311, 311, 50, 3, 1, 1, 4, 155, 1954, 4101, 1954, 155, 4, 1, 1, 5, 508, 11914, 56864, 56864, 11914, 508, 5, 1, 1, 6, 1343, 76003, 728857, 1532496, 728857, 76003, 1343, 6, 1

Offset: 0

Alois P. Heinz, Nov 25 2016

			A(2,3) = A(3,2) = 7:
  .___.  .___.  .___.  .___.  .___.  .___.  .___.
  |_|_|  |   |  |   |  | |_|  |_| |  | ._|  |_. |
  |_|_|  | ._|  |_. |  |   |  |   |  | |_|  |_| |
  |_|_|  |_|_|  |_|_|  |___|  |___|  |___|  |___| .
.
Square array A(n,k) begins:
  1, 1,   1,     1,      1,        1,          1, ...
  1, 1,   1,     1,      1,        2,          3, ...
  1, 1,   1,     7,     25,       50,        155, ...
  1, 1,   7,    50,    311,     1954,      11914, ...
  1, 1,  25,   311,   4101,    56864,     728857, ...
  1, 2,  50,  1954,  56864,  1532496,   42238426, ...
  1, 3, 155, 11914, 728857, 42238426, 2492016728, ...

Liang Kai, Antidiagonals n = 0..23, flattened (Antidiagonals n = 0..15 from Alois P. Heinz)
Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
Wikipedia, Pentomino

Columns (or rows) k=0-7 give: A000012, A003520, A278874, A278875, A278876, A278456, A278877, A278878.

Cf. A174249, A233427.

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

Alois P. Heinz, Dec 23 2013

nonn

			a(4) = 4:
._______.  ._______.  ._______.  ._______.
|_. |_. |  | ._| ._|  |_. | ._|  | ._|_. |
| | | | |  | | | | |  | | | | |  | | | | |
| | | | |  | | | | |  | | | | |  | | | | |
| |_| |_|  |_| |_| |  | |_|_| |  |_| | |_|
|___|___|  |___|___|  |___|___|  |___|___|.
a(5) = 2:
._________.  ._________.
| | ._____|  |_____. | |
| |_| |_. |  | ._| |_| |
| |_. ._| |  | |_. ._| |
|___|_| | |  | | |_|___|
|_______|_|  |_|_______|.

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

Cf. A077909, A174249, A233427, A234931, A247125, A264812.

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 *)

G.f.: -1/(2*x^5+2*x^2-1).

a(n) = 2*(a(n-2)+a(n-5)) for n>4, a(1)=a(3)=0, a(0)=1, a(2)=2, a(4)=4.

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

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

cp's OEIS Frontend

A174249 Number of tilings of a 5 X n rectangle with n pentominoes of any shape.

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A099390 Array T(m,n) read by antidiagonals: number of domino tilings (or dimer tilings) of the m X n grid (or m X n rectangle), for m>=1, n>=1.

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A230031 Number A(n,k) of tilings of a k X n rectangle using tetrominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A233320 Number A(n,k) of tilings of a k X n rectangle using trominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Formula

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

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A278657 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape and monominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A234312 Number of tilings of a 5 X n rectangle using n pentominoes of shapes L, X.

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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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