A236578 -id:A236578 - cp's OEIS frontend

A236576 The number of tilings of a 5 X (3n) floor with 1 X 3 trominoes.

1, 4, 22, 121, 664, 3643, 19987, 109657, 601624, 3300760, 18109345, 99355414, 545105209, 2990674357, 16408085929, 90021597712, 493896002842, 2709719309845, 14866649448256, 81564634762843, 447497579542135

Offset: 0

R. J. Mathar, Jan 29 2014

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mudit Aggarwal and Samrith Ram, Generating functions for straight polyomino tilings of narrow rectangles, arXiv:2206.04437 [math.CO], 2022.
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 20.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (6,-3,1).

Cf. A000930 (3 X n floor), A049086 (4 X 3n floor), A236577, A236578.

g := (1-x)^2/(1-6*x+3*x^2-x^3) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

CoefficientList[Series[(1 - x)^2/(1 - 6 x + 3 x^2 - x^3), {x,0,50}], x] (* G. C. Greubel, Apr 29 2017 *)
LinearRecurrence[{6, -3, 1}, {1, 4, 22}, 30] (* M. Poyraz Torcuk, Nov 06 2021 *)

my(x='x+O('x^50)); Vec((1-x)^2/(1-6*x+3*x^2-x^3)) \\ G. C. Greubel, Apr 29 2017

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

a(n) = 6*a(n-1) - 3*a(n-2) + a(n-3). - M. Poyraz Torcuk, Oct 24 2021

A236577 The number of tilings of a 6 X n floor with 1 X 3 trominoes.

Original entry on oeis.org

1, 1, 1, 6, 13, 22, 64, 155, 321, 783, 1888, 4233, 9912, 23494, 54177, 126019, 295681, 687690, 1600185, 3738332, 8712992, 20293761, 47337405, 110368563, 257206012, 599684007, 1398149988, 3259051800, 7597720649, 17712981963

Offset: 0

R. J. Mathar, Jan 29 2014

Tilings are counted irrespective of internal symmetry: Tilings that match each other after rotations and/or reflections are counted with their multiplicity.

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 21.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq (14).
Index entries for linear recurrences with constant coefficients, signature (1,1,7,-1,-5,-10,-1,3,5,1,-1,-1).

Cf. A000930 (3Xn floor), A049086 (4X3n floor), A236576 - A236578.
Column k=3 of A250662.
Cf. A251073.

g := (1-x^3)^2*(-x^2+1-x^3)/ (-x^10+x^12+x^11+10*x^6-5*x^9-3*x^8+x^7+x^4-7*x^3+5*x^5-x^2-x+1) ;
taylor(%,x=0,30) ;
gfun[seriestolist](%) ;

CoefficientList[Series[(1 - x^3)^2*(-x^2 + 1 - x^3)/(-x^10 + x^12 + x^11 + 10*x^6 - 5*x^9 - 3*x^8 + x^7 + x^4 - 7*x^3 + 5*x^5 - x^2 - x + 1), {x, 0, 50}], x] (* G. C. Greubel, Apr 27 2017 *)

x='x+O('x^50); Vec((1-x^3)^2*(-x^2+1-x^3)/(-x^10+x^12+x^11+10*x^6 -5*x^9-3*x^8+x^7+x^4-7*x^3+5*x^5-x^2-x+1)) \\ G. C. Greubel, Apr 27 2017

G.f.: See the definition of g in the Maple code.

A351324 Number of tilings of a 7 X 3n rectangle with right trominoes.

Original entry on oeis.org

1, 0, 520, 22656, 1795360, 115363072, 7876120608, 527256809600, 35522814546496, 2388257605782016, 160678147466414272, 10807663334085120512, 727010169682181839360, 48903265220016072792320, 3289569236212332037229184, 221278350342281369716796672

Offset: 0

Gerhard Kirchner, Feb 21 2022

See A351322 for algorithm.

This is the Hadamard sum of the following 4 sequences: 0, 0,0,0, 158208,.. (tilings which have both vertical and horizontal faults), 0,0,480,6144, 125952 ... (tilings which have horizontal faults but no vertical faults), 00,0,0,112192,.. (tilings which have vertical but no horizontal faults), 1, 0,40, 16512, 1399008 ,... (tilings which have neither horizontal nor vertical faults). - R. J. Mathar, Dec 08 2022

Cf. A077957, A000079, A046984, A084478, A351322, A351323, A236578 (straight trominoes), A233343 (mixed trominoes).

G.f.: (1 - 22*x - 1831*x^2 - 29454*x^3 - 270630*x^4 - 2070388*x^5 - 12125943*x^6 - 48147976*x^7 - 151548064*x^8 - 417242784*x^9 - 423562924*x^10 + 586224672*x^11 + 915719344*x^12 + 349980800*x^13 + 371621248*x^14 - 6541312*x^15 - 9691136*x^16 + 589824*x^17)/(1 - 22*x - 2351*x^2 - 40670*x^3 - 345038*x^4 - 3522884*x^5 - 28528327*x^6 - 145350120*x^7 - 623982088*x^8 - 2110011040*x^9 - 1354478796*x^10 + 9281598624*x^11 + 15001687984*x^12 + 3456230016*x^13 - 3194643904*x^14 - 1637793792*x^15 - 575934464*x^16 + 65175552*x^17).

a(n) = 22*a(n-1) + 2351*a(n-2) + 40670*a(n-3) + 345038*a(n-4) + 3522884*a(n-5) + 28528327*a(n-6) + 145350120*a(n-7) + 623982088*a(n-8) + 2110011040*a(n-9) + 1354478796*a(n-10) - 9281598624*a(n-11) - 15001687984*a(n-12) - 3456230016*a(n-13) + 3194643904*a(n-14) + 1637793792*a(n-15) + 575934464*a(n-16) - 65175552*a(n-17) for n>16.

cp's OEIS Frontend

A236576 The number of tilings of a 5 X (3n) floor with 1 X 3 trominoes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A236577 The number of tilings of a 6 X n floor with 1 X 3 trominoes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A351324 Number of tilings of a 7 X 3n rectangle with right trominoes.

Views

Author

Keywords

Comments

Crossrefs

Formula