A134438 -id:A134438 - 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

A174248 Number of tilings of a 4 X n rectangle with n tetrominoes of any shape.

Original entry on oeis.org

1, 1, 4, 23, 117, 454, 2003, 9157, 40899, 179399, 796558, 3546996, 15747348, 69834517, 310058192, 1376868145, 6112247118, 27132236455, 120453362938, 534754586459, 2373975139658, 10538953415410, 46786795734201, 207705902269424, 922089495910044, 4093525019450760

Offset: 0

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
S. Butler, J. Ekstrand, and S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013.
R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings.
Wikipedia, Tetris.
Wikipedia, Tetromino.
Index entries for linear recurrences with constant coefficients, signature (2, 8, 8, 54, -77, -290, -76, -548, 469, 2258, 414, 1970, -1053, -6885, -1620, -3349, 1102, 9566, 3210, 2786, -489, -6047, -2600, -1102, 60, 1476, 659, 225, 39, -123, -50, -13, -3, 3, 1).

Cf. A134438, A174249, A226322, A232497, A232684, A232698, A232722, A233191, A233266.

Column k=4 of A230031.

G.f.: -(x^31 +3*x^30 -2*x^29 -7*x^28 -25*x^27 -78*x^26 +23*x^25 +116*x^24 +217*x^23 +604*x^22 -21*x^21 -556*x^20 -649*x^19 -1621*x^18 -175*x^17 +727*x^16 +523*x^15 +1707*x^14 +236*x^13 -470*x^12 -143*x^11 -749*x^10 -133*x^9 +166*x^8 +15*x^7 +126*x^6 +27*x^5 -23*x^4 -x^3 -6*x^2 -x +1) / (x^35 +3*x^34 -3*x^33 -13*x^32 -50*x^31 -123*x^30 +39*x^29 +225*x^28 +659*x^27 +1476*x^26 +60*x^25 -1102*x^24 -2600*x^23 -6047*x^22 -489*x^21 +2786*x^20 +3210*x^19 +9566*x^18 +1102*x^17 -3349*x^16 -1620*x^15 -6885*x^14 -1053*x^13 +1970*x^12 +414*x^11 +2258*x^10 +469*x^9 -548*x^8 -76*x^7 -290*x^6 -77*x^5 +54*x^4 +8*x^3 +8*x^2 +2*x -1). - Alois P. Heinz, Nov 26 2013

a(0) inserted, a(11)-a(22) from Alois P. Heinz, May 07 2013

a(23)-a(25) from Alois P. Heinz, Nov 26 2013

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.

A215826 Number of ways in which a 9 X n grid can be tiled with trominoes.

Original entry on oeis.org

1, 1, 41, 3127, 41813, 1269900, 45832761, 1064557805, 30860212081, 928789262080, 25020222581494, 714819627084057, 20574308184277971, 576115800837801057, 16381774291037991059, 466431115279461257920, 13190758349044182698371, 374524994697062170913555

Offset: 0

V. Raman, Aug 23 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..700 (terms n = 1..42 from V. Raman)
Project Euler, Problem 161: Triominoes.

Cf. A215827 (number of memoizations needed to calculate a(n)), A134438, A233290, A269664.

Column k=9 of A233320.

A233290 Number of tilings of a 6 X n rectangle with 2n trominoes of any shape.

Original entry on oeis.org

1, 1, 11, 170, 939, 8342, 80092, 614581, 5271923, 45832761, 379419508, 3223419241, 27438555522, 231096250072, 1958024834151, 16593169804557, 140295718998907, 1187830239559588, 10056816580083721, 85104482994177208, 720410915948382970, 6098207777286812381

Offset: 0

Alois P. Heinz, Dec 06 2013

nonn

			a(2) = 11:
.___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
| | | | ._| |_. | | ._| | | | |_. | | | | |_. | | ._| | ._| |_. |
| | | |_| | | |_| |_| | | | | | |_| | | | | |_| |_| | |_| | | |_|
|_|_| |___| |___| |___| |_|_| |___| |_|_| |___| |___| | | | | | |
| | | | ._| |_. | | | | | ._| | | | |_. | | ._| |_. | | |_| |_| |
| | | |_| | | |_| | | | |_| | | | | | |_| |_| | | |_| |_| | | |_|
|_|_| |___| |___| |_|_| |___| |_|_| |___| |___| |___| |___| |___|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Recurrence (of order 126) and G.f.
Wikipedia, Tromino

Column k=6 of A233320.

Cf. A134438, A215826, A269664.

A174250 Number of tilings of a 6 X n rectangle with n hexominoes of any shape.

Original entry on oeis.org

1, 1, 6, 132, 2369, 33344, 451206, 5850115, 81459922, 1144259389, 15946621499

Offset: 0

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

R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings
Wikipedia, Hexomino

Cf. A134438, A174248, A174249, A174251, A174252, A174253.

A174251 Number of tilings of a 7 X n rectangle with n heptominoes of any shape.

Original entry on oeis.org

1, 1, 7, 259, 9525, 270827, 6633399, 158753814, 3825111851, 96608374284, 2446223788303

Offset: 0

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

R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings

Cf. A134438, A174248, A174249, A174250, A174252, A174253.

A174252 Number of tilings of an 8 X n rectangle with n octominoes of any shape.

Original entry on oeis.org

1, 1, 8, 546, 39731, 2152050, 99697633, 4292655082, 187497290034, 8378760802160, 385296986628990

Offset: 0

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

R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings

Cf. A134438, A174248, A174249, A174250, A174251, A174253.

A174253 Number of tilings of a 9 X n rectangle with n nonominoes of any shape.

Original entry on oeis.org

1, 1, 9, 1095, 145415, 15661597, 1418159011, 112976454947, 8812020831683, 706152947468301, 58015563977931125

Offset: 0

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

R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings

Cf. A134438, A174248, A174249, A174250, A174251, A174252.

A269664 Number of tilings of a 12 X n rectangle with 4n trominoes of any shape.

Original entry on oeis.org

1, 1, 153, 58234, 1895145, 198253934, 27438555522, 1949314526229, 193553900967497, 20574308184277971, 1830607857363940042, 178792253082742021463, 17735061025562799941630, 1679378707647721857218932, 163105210594579645492072521, 15894545877032388610890500803

Offset: 0

Alois P. Heinz, Mar 02 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Wikipedia, Tromino

Column k=12 of A233320.

Cf. A134438, A215826, A233290.

cp's OEIS Frontend

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

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A174248 Number of tilings of a 4 X n rectangle with n tetrominoes of any shape.

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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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A215826 Number of ways in which a 9 X n grid can be tiled with trominoes.

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A233290 Number of tilings of a 6 X n rectangle with 2n trominoes of any shape.

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A174250 Number of tilings of a 6 X n rectangle with n hexominoes of any shape.

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A174251 Number of tilings of a 7 X n rectangle with n heptominoes of any shape.

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A174252 Number of tilings of an 8 X n rectangle with n octominoes of any shape.

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A174253 Number of tilings of a 9 X n rectangle with n nonominoes of any shape.

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A269664 Number of tilings of a 12 X n rectangle with 4n trominoes of any shape.

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