A270061 -id:A270061 - cp's OEIS frontend

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.

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.

A270071 Number of tilings of an n X n square using monominoes and trominoes of any shape.

Original entry on oeis.org

1, 1, 5, 93, 7517, 2501946, 3473827027, 19823918037604, 466326405037190729, 45283517791861200639757, 18137923612752525595542410804, 29967243098804386765778629249653298, 204247259656641635611144023845458294818885

Offset: 0

Alois P. Heinz, Mar 09 2016

nonn

Liang Kai, Table of n, a(n) for n = 0..17
Wikipedia, Tromino

Main diagonal of A270061.

A270063 Number of tilings of a 3 X n rectangle using monominoes and trominoes of any shape.

Original entry on oeis.org

1, 2, 14, 93, 590, 3710, 23509, 148796, 940916, 5952243, 37655502, 238204978, 1506870209, 9532433288, 60301864000, 381467462217, 2413150656124, 15265511608858, 96569119066667, 610893045560410, 3864489168524208, 24446630393071735, 154648573589573678

Offset: 0

Alois P. Heinz, Mar 09 2016

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

Column k=3 of A270061.

Cf. A134438.

G.f.: -(x^15 +2*x^14 +5*x^13 -4*x^12 -7*x^11 -21*x^10 +12*x^9 +11*x^8 +24*x^7 -21*x^6 -13*x^5 +16*x^3 +2*x^2 +2*x -1) / (x^18 +2*x^17 +5*x^16 -9*x^15 -21*x^14 -54*x^13 +32*x^12 +46*x^11 +147*x^10 +21*x^9 -40*x^8 -110*x^7 -17*x^6 -11*x^5 -32*x^4 -37*x^3 -8*x^2 -4*x +1).

A270062 Number of tilings of a 2 X n rectangle using monominoes and trominoes of any shape.

Original entry on oeis.org

1, 1, 5, 14, 45, 140, 438, 1371, 4287, 13413, 41956, 131249, 410572, 1284352, 4017713, 12568213, 39315905, 122988066, 384731445, 1203517448, 3764844982, 11777193395, 36841433019, 115247422841, 360517151000, 1127770261265, 3527892525112, 11035958382864

Offset: 0

Alois P. Heinz, Mar 09 2016

			a(3) = 14:
._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
|_____|  |_|_|_|  |_____|  |_| |_|  | |_|_|  | ._|_|  |_. |_|
|_____|  |_____|  |_|_|_|  |___|_|  |___|_|  |_|_|_|  |_|_|_|
.
._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
|_|_|_|  | ._| |  | |_. |  |_|_| |  |_| |_|  |_| ._|  |_|_. |
|_|_|_|  |_|___|  |___|_|  |_|___|  |_|___|  |_|_|_|  |_|_|_|  .
.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Tromino
Index entries for linear recurrences with constant coefficients, signature (1,5,5,1,-1,-1).

Column k=2 of A270061.

a:= n-> (Matrix(6, (i, j)-> `if`(i+1=j, 1, `if`(i=6, [-1$2, 1,
         5$2, 1][j], 0)))^n. <<1, 1, 5, 14, 45, 140>>)[1, 1]:
seq(a(n), n=0..30);

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

a(n) = a(n-1) +5*(a(n-2)+a(n-3)) +a(n-4) -a(n-5) -a(n-6).

A270064 Number of tilings of a 4 X n rectangle using monominoes and trominoes of any shape.

Original entry on oeis.org

1, 3, 45, 590, 7517, 96176, 1238818, 15913695, 204383373, 2626057921, 33738191906, 433438712623, 5568545948434, 71541033899738, 919110571684493, 11808116466700575, 151702772808195855, 1948975423808006804, 25039128916718657873, 321685935180001027054

Offset: 0

Alois P. Heinz, Mar 09 2016

Alois P. Heinz, Table of n, a(n) for n = 0..900
Wikipedia, Tromino

Column k=4 of A270061.

G.f.: -(x^42 -2*x^41 +3*x^40 -45*x^39 +75*x^38 -104*x^37 +788*x^36 -1328*x^35 +1326*x^34 -8174*x^33 +13318*x^32 -9055*x^31 +51980*x^30 -68274*x^29 +18138*x^28 -163965*x^27 +144209*x^26 +50099*x^25 +307359*x^24 -237066*x^23 -88998*x^22 -420220*x^21 +288839*x^20
+56875*x^19 +263379*x^18 -179966*x^17 +19258*x^16 -47380*x^15 +99349*x^14 -28851*x^13 -16616*x^12 -21781*x^11 +15041*x^10 -705*x^9 -2136*x^8 -1268*x^7 +2193*x^6 +558*x^5 -149*x^4 -187*x^3 -25*x^2 -4*x +1)
/ (x^45 -2*x^44 +4*x^43 -58*x^42 +90*x^41 -162*x^40 +1072*x^39 -1745*x^38 +1906*x^37 -10656*x^36 +21746*x^35 -17639*x^34 +90014*x^33 -190035*x^32 +136443*x^31 -516421*x^30 +828239*x^29 -326624*x^28 +1290743*x^27 -1284720*x^26 -519116*x^25 -1599589*x^24 +1444541*x^23 +956723*x^22
+2264534*x^21 -1759452*x^20 -691070*x^19 -1308065*x^18 +953491*x^17 +192415*x^16 +204107*x^15 -475505*x^14 +77942*x^13 +133756*x^12 +68320*x^11 -73312*x^10 -9470*x^9 +10934*x^8 +4282*x^7 -4428*x^6 -1244*x^5 +386*x^4 +315*x^3 +49*x^2 +7*x -1).

A270065 Number of tilings of a 5 X n rectangle using monominoes and trominoes of any shape.

Original entry on oeis.org

1, 4, 140, 3710, 96176, 2501946, 65410388, 1707101599, 44538604036, 1162378878394, 30334909910156, 791636651386694, 20659194865202206, 539139829390310029, 14069828639020139191, 367177676400256677129, 9582168262012437486944, 250064069442551102774366

Offset: 0

Alois P. Heinz, Mar 09 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..700
Vaclav Kotesovec, Recurrence (of order 135) and g.f.
Wikipedia, Tromino

Column k=5 of A270061.

A270066 Number of tilings of a 6 X n rectangle using monominoes and trominoes of any shape.

Original entry on oeis.org

1, 6, 438, 23509, 1238818, 65410388, 3473827027, 184140417890, 9756894398324, 517174361985073, 27412100990502042, 1452889626133073024, 77006768282516987969, 4081555197498259422846, 216332403430110823588762, 11466150014825389043509015

Offset: 0

Alois P. Heinz, Mar 09 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..580
Wikipedia, Tromino

Column k=6 of A270061.

A270067 Number of tilings of a 7 X n rectangle using monominoes and trominoes of any shape.

Original entry on oeis.org

1, 9, 1371, 148796, 15913695, 1707101599, 184140417890, 19823918037604, 2133436326533143, 229681994753502100, 24725866354382952217, 2661724100255128134369, 286536699674194768262220, 30845921492404301091244774, 3320584102598070051873302366

Offset: 0

Alois P. Heinz, Mar 09 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Tromino

Column k=7 of A270061.

A270068 Number of tilings of an 8 X n rectangle using monominoes and trominoes of any shape.

Original entry on oeis.org

1, 13, 4287, 940916, 204383373, 44538604036, 9756894398324, 2133436326533143, 466326405037190729, 101965922336345576219, 22294594753523459270694, 4874505512330163994345181, 1065778806957316983452583926, 233025803338941141247495062594

Offset: 0

Alois P. Heinz, Mar 09 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..430
Wikipedia, Tromino

Column k=8 of A270061.

A270069 Number of tilings of a 9 X n rectangle using monominoes and trominoes of any shape.

Original entry on oeis.org

1, 19, 13413, 5952243, 2626057921, 1162378878394, 517174361985073, 229681994753502100, 101965922336345576219, 45283517791861200639757, 20109634299824273864511862, 8930062817269356408791188106, 3965614916098256915340694883017, 1761030770921730133912679646624228

Offset: 0

Alois P. Heinz, Mar 09 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Tromino

Column k=9 of A270061.

cp's OEIS Frontend

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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A270071 Number of tilings of an n X n square using monominoes and trominoes of any shape.

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A270063 Number of tilings of a 3 X n rectangle using monominoes and trominoes of any shape.

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A270062 Number of tilings of a 2 X n rectangle using monominoes and trominoes of any shape.

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A270064 Number of tilings of a 4 X n rectangle using monominoes and trominoes of any shape.

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A270065 Number of tilings of a 5 X n rectangle using monominoes and trominoes of any shape.

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A270066 Number of tilings of a 6 X n rectangle using monominoes and trominoes of any shape.

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A270067 Number of tilings of a 7 X n rectangle using monominoes and trominoes of any shape.

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A270068 Number of tilings of an 8 X n rectangle using monominoes and trominoes of any shape.

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A270069 Number of tilings of a 9 X n rectangle using monominoes and trominoes of any shape.

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