A236576 -id:A236576 - cp's OEIS frontend

A084478 Number of tilings of a 5 X 3n rectangle with right trominoes.

1, 0, 72, 384, 8544, 76800, 1168512, 12785664, 170678784, 2014648320, 25633231872, 311423852544, 3892030055424, 47803588208640, 593425578949632, 7318730222874624, 90624271197041664, 1119402280975349760, 13847850677651745792, 171150049715628539904

Offset: 0

Ralf Stephan, May 27 2003

A right tromino is a 3-celled L-shaped piece (a 2 X 2 square with one of the four cells omitted). - N. J. A. Sloane, Mar 28 2017
There is a sign typo with respect to the g.f. in the paper.
The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 2048, 0, 65536, 0,.. (tilings which have both vertical and horizontal faults), 0, 0, 64, 0, 0, 0, 0, 0.. (tilings which have horizontal but no vertical faults), 0, 0, 0, 0, 3136, 55296, 939008, 11649024... (tilings which have vertical faults but no horizontal faults), .. 1, 0, 8, 384, 3360, 21504, 163968 (essentially A084479) which have neither vertical nor horizontal faults. - R. J. Mathar, Dec 08 2022

Colin Barker, Table of n, a(n) for n = 0..900
D. Merlini, R. Sprugnoli, M. C. Verri, Strip tiling and regular grammars, Theo. Comp. Sci. 242 (1-2) (2000) 109-124, Proof of Theorem 4.2 (typo t^5 in the denominator of g.f. ought be t^6)
C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
Index entries for linear recurrences with constant coefficients, signature (2,103,280,380).

Cf. A046984, A084477, A084479 (INVERT transform), A084480, A084481,A351323, A351324, A236576 (straight trominoes), A233340 (mixed trominoes).

LinearRecurrence[{2, 103, 280, 380}, {72, 384, 8544, 76800}, 20] (* Jean-François Alcover, Jan 07 2019 *)

Vec(24*x^2*(3 + 10*x + 15*x^2) / (1 - 2*x - 103*x^2 - 280*x^3 - 380*x^4) + O(x^30)) \\ Colin Barker, Mar 27 2017

G.f.: (1 - 2*z - 31*z^2 - 40*z^3 - 20*z^4) / (1 - 2*z - 103*z^2 - 280*z^3 - 380*z^4).

a(n) = 2*a(n-1) + 103*a(n-2) + 280*a(n-3) + 380*a(n-4) for n > 4. - Colin Barker, Mar 27 2017

a(0) and a(1) prepended by Alois P. Heinz, Feb 21 2022

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.

A236578 The number of tilings of a 7 X (3n) floor with 1 X 3 trominoes.

Original entry on oeis.org

1, 9, 155, 2861, 52817, 972557, 17892281, 329097125, 6052932495, 111328274273, 2047599783121, 37660384283749, 692666924307063, 12739845501187821, 234317040993180833, 4309665744385061493, 79265335342431559977

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..785
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 22.
R. J. Mathar, Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices, arXiv:1406.7788 [math.CO], 2014, eq. (15).

Cf. A000930 (3Xn floor), A049086 (4X3n floor), A236576, A236577.

p := (x-1)^2*(-x^15 +14*x^14 -104*x^13 +527*x^12 -1971*x^11 +5573*x^10 -11973*x^9 +19465*x^8 -23695*x^7 +21166*x^6 -13512*x^5 +5915*x^4 -1685*x^3 +291*x^2 -27*x+1) ;
q := -17*x^17 +293180*x^8 -236178*x^7 +142400*x^6 -62621*x^5 +19420*x^4 -4062*x^3 +533*x^2 -38*x +x^18 +1 +151*x^16 -946*x^15 +4558*x^14 -17135*x^13 +50164*x^12 -114198*x^11 +202080*x^10 -277277*x^9 ;
taylor(p/q,x=0,30) ;
gfun[seriestolist](%) ;

G.f.: p(x)/q(x) with polynomials p and q defined in the Maple code.

A351323 Number of tilings of a 6 X n rectangle with right trominoes.

Original entry on oeis.org

1, 0, 4, 8, 18, 72, 162, 520, 1514, 4312, 13242, 39088, 118586, 361712, 1103946, 3403624, 10513130, 32614696, 101530170, 316770752, 990771834, 3104283168, 9741133578, 30606719000, 96263812906, 303028237848, 954563802106, 3008665176560, 9487377712634, 29928407213328

Offset: 0

Gerhard Kirchner, Feb 21 2022

nonn

See A351322 for algorithm. The subsequence 1,8,162,... for 6 X 3n rectangles also has a depending recurrence with 11 parameters.

The sequence is the Hadamard sum of the following 4 sequences: 0, 0, 0, 0, 16, 0, 128, 0, 256, 768, 1024,0, 13440, 0, 16384, .. (tilings which have both horizontal and vertical faults), 0, 0, 4, 8, 0, 0, 16, 0, 0, 128, 0, 0, 1536, 0, 0,.. (tilings which have horizontal faults but no vertical faults), 0, 0, 0, 0, 0, 64, 16, 480, 1140, 3200, 11208, 36032, 95924, 333856, 1003096,.. (tilings which have vertical faults but no horizontal faults), 1, 0, 0, 0, 2, 8, 2, 40, 118, 216, 1010, 3056, 7686, 27856, 84466,... (tilings which have neither vertical nor horizontal faults). - R. J. Mathar, Dec 08 2022

			For a 6 X 2 rectangle there are 4 tilings:
   ___   ___   ___   ___
  |  _| |  _| |_  | |_  |
  |_| | |_| | | |_| | |_|
  |___| |___| |___| |___|
  |  _| |_  | |  _| |_  |
  |_| | | |_| |_| | | |_|
  |___| |___| |___| |___|

Index entries for linear recurrences with constant coefficients, signature (2,8,2,-43,-42,36,102,0,-44,-8,-8).

Cf. A077957, A000079, A046984, A084478, A351322, A351324, A236576 (straight trominoes), A233290 (mixed trominoes).

G.f.: (1 - x)*(1 - x - 5*x^2 - 7*x^3 + 6*x^4 + 12*x^5 + 6*x^6)/(1 - 2*x - 8*x^2 - 2*x^3 + 43*x^4 + 42*x^5 - 36*x^6 - 102*x^7 + 44*x^9 + 8*x^10 + 8*x^11).

a(n) = Sum_{i=0..10} b(i)*a(n-11+i) for n>10 where {b(i)} = {-8,-8,-44,0,102,36,-42,-43,2,8,2}.

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A084478 Number of tilings of a 5 X 3n rectangle with right trominoes.

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

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A236578 The number of tilings of a 7 X (3n) floor with 1 X 3 trominoes.

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A351323 Number of tilings of a 6 X n rectangle with right trominoes.

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