A054854 -id:A054854 - cp's OEIS frontend

A063443 Number of ways to tile an n X n square with 1 X 1 and 2 X 2 tiles.

1, 1, 2, 5, 35, 314, 6427, 202841, 12727570, 1355115601, 269718819131, 94707789944544, 60711713670028729, 69645620389200894313, 144633664064386054815370, 540156683236043677756331721, 3641548665525780178990584908643, 44222017282082621251230960522832336

Offset: 0

Reiner Martin, Jul 23 2001

a(n) is also the number of ways to populate an n-1 X n-1 chessboard with nonattacking kings (including the case of zero kings). Cf. A193580. - Andrew Woods, Aug 27 2011

Also the number of vertex covers and independent vertex sets of the n-1 X n-1 king graph.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 343

Andrew Woods and Vaclav Kotesovec and Johan Nilsson, Table of n, a(n) for n = 0..40 (terms 0..21 from Andrew Woods, terms 22..24 from Vaclav Kotesovec and terms 25..40 from Johan Nilsson)
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 68-69.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO], 2016, Section 4.1.
J. Nilsson, On Counting the Number of Tilings of a Rectangle with Squares of Size 1 and 2, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.2.
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A001045, A006506, A054854, A054855, A063650-A063653, A067966, etc.
Cf. A045846, A211348, A247413, A201513.
Cf. A212269, A067958.
a(n) = row sum n-1 of A193580.
Main diagonal of A245013.

Needs["LinearAlgebra`MatrixManipulation`"] Remove[mat] step[sa[rules1_, {dim1_, dim1_}], sa[rules2_, {dim2_, dim2_}]] := sa[Join[rules2, rules1 /. {x_Integer, y_Integer} -> {x + dim2, y}, rules1 /. {x_Integer, y_Integer} -> {x, y + dim2}], {dim1 + dim2, dim1 + dim2}] mat[0] = sa[{{1, 1} -> 1}, {1, 1}]; mat[1] = sa[{{1, 1} -> 1, {1, 2} -> 1, {2, 1} -> 1}, {2, 2}]; mat[n_] := mat[n] = step[mat[n - 2], mat[n - 1]]; A[n_] := mat[n] /. sa -> SparseArray; F[n_] := MatrixPower[A[n], n + 1][[1, 1]]; (* Mark McClure (mcmcclur(AT)bulldog.unca.edu), Mar 19 2006 *)
$RecursionLimit = 1000; Clear[a, b]; b[n_, l_List] := b[n, l] = Module[{m=Min[l], k}, If[m>0, b[n-m, l-m], If[n == 0, 1, k=Position[l, 0, 1, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[n>1 && k 2, k+1 -> 2}]], 0]]]]; a[n_] := a[n] = If[n<2, 1, b[n, Table[0, {n}]]]; Table[Print[a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 11 2014, after Alois P. Heinz *)

Lim_{n -> infinity} (a(n))^(1/n^2) = A247413 = 1.342643951124... . - Brendan McKay, 1996

4 more terms from R. H. Hardin, Jan 23 2002
2 more terms from Keith Schneider (kschneid(AT)bulldog.unca.edu), Mar 19 2006
5 more terms from Andrew Woods, Aug 27 2011
a(22)-a(24) in b-file from Vaclav Kotesovec, May 01 2012
a(0) inserted by Alois P. Heinz, Sep 17 2014
a(25)-a(40) in b-file from Johan Nilsson, Mar 10 2016

A245013 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and 2 X 2 squares; 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, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 5, 5, 1, 1, 1, 1, 8, 11, 11, 8, 1, 1, 1, 1, 13, 21, 35, 21, 13, 1, 1, 1, 1, 21, 43, 93, 93, 43, 21, 1, 1, 1, 1, 34, 85, 269, 314, 269, 85, 34, 1, 1, 1, 1, 55, 171, 747, 1213, 1213, 747, 171, 55, 1, 1

Offset: 0

Alois P. Heinz, Sep 16 2014

			A(3,3) = 5:
  ._._._.  .___._.  ._.___.  ._._._.  ._._._.
  |_|_|_|  |   |_|  |_|   |  |_|_|_|  |_|_|_|
  |_|_|_|  |___|_|  |_|___|  |_|   |  |   |_|
  |_|_|_|  |_|_|_|  |_|_|_|  |_|___|  |___|_| .
Square array A(n,k) begins:
  1, 1,  1,  1,   1,    1,     1,      1, ...
  1, 1,  1,  1,   1,    1,     1,      1, ...
  1, 1,  2,  3,   5,    8,    13,     21, ...
  1, 1,  3,  5,  11,   21,    43,     85, ...
  1, 1,  5, 11,  35,   93,   269,    747, ...
  1, 1,  8, 21,  93,  314,  1213,   4375, ...
  1, 1, 13, 43, 269, 1213,  6427,  31387, ...
  1, 1, 21, 85, 747, 4375, 31387, 202841, ...

Liang Kai, Antidiagonals n = 0..80, flattened (antidiagonals n = 0..45 from Alois P. Heinz)
Kai Liang, Independent Set Enumeration in King Graphs by Tensor Network Contractions, arXiv:2505.12776 [math.CO], 2025. See p. 4.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO], 2016.
Johan Nilsson, On Counting the Number of Tilings of a Rectangle with Squares of Size 1 and 2, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.2.

Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A001045(n+1), A054854, A054855, A063650, A063651, A063652, A063653, A063654.

Main diagonal gives A063443.

b:= proc(n, l) option remember; local m, k; m:= min(l[]);
      if m>0 then b(n-m, map(x->x-m, l))
    elif n=0 then 1
    else for k while l[k]>0 do od; b(n, subsop(k=1, l))+
         `if`(n>1 and k `if`(min(n, k)<2, 1, b(max(n, k), [0$min(n, k)])):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, l_] := b[n, l] = Module[{m=Min[l], k}, If[m>0, b[n-m, l-m], If[n == 0, 1, k=Position[l, 0, 1, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[n>1 && k 2, k+1 -> 2}]], 0]]]]; A[n_, k_] := If[Min[n, k]<2, 1, b[Max[n, k], Table[0, {Min[n, k]}]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2014, after Alois P. Heinz *)

A226322 Number of tilings of a 4 X n rectangle using L tetrominoes and 2 X 2 tiles.

Original entry on oeis.org

1, 0, 3, 6, 19, 48, 141, 378, 1063, 2920, 8115, 22418, 62123, 171876, 475919, 1317250, 3646681, 10094356, 27943739, 77353070, 214129845, 592752572, 1640859689, 4542223926, 12573787053, 34806745800, 96352029241, 266721635838, 738338745535, 2043868995512

Offset: 0

Alois P. Heinz, Jun 03 2013

			a(3) = 6:
._____.  ._____.  .___._.  ._.___.  ._____.  ._____.
| .___|  |___. |  |   | |  | |   |  |___. |  | .___|
|_|_. |  | ._|_|  |___| |  | |___|  |   |_|  |_|   |
|   | |  | |   |  | |___|  |___| |  |___| |  | |___|
|___|_|  |_|___|  |_____|  |_____|  |_____|  |_____|

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.
Index entries for linear recurrences with constant coefficients, signature (0,5,6,4,0,-1,0,-3,-2,-4,0,-2).

Cf. A054854, A054856, A084480, A165716, A165791, A165799, A174248, A232497, A233191, A233266.

a:= n-> (Matrix(12, (i, j)-> `if`(i+1=j, 1, `if`(i=12,
    [-2, 0, -4, -2, -3, 0, -1, 0, 4, 6, 5, 0][j], 0)))^(n+8).
    <<-1, 0, 1/2, [0$5][], 1, 0, 3, 6>>)[1, 1]:
seq(a(n), n=0..40);

a[n_] := MatrixPower[ Table[ If[i+1 == j, 1, If[i == 12, {-2, 0, -4, -2, -3, 0, -1, 0, 4, 6, 5, 0}[[j]], 0]], {i, 1, 12}, {j, 1, 12}], n+8].{-1, 0, 1/2, 0, 0, 0, 0, 0, 1, 0, 3, 6} // First; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 05 2013, after Maple *)

G.f.: (x^6+2*x^2-1) / (-2*x^12 -4*x^10 -2*x^9 -3*x^8 -x^6 +4*x^4 +6*x^3 +5*x^2-1).

A054855 Number of ways to tile a 5 X n area with 1 X 1 and 2 X 2 tiles.

Original entry on oeis.org

1, 1, 8, 21, 93, 314, 1213, 4375, 16334, 59925, 221799, 817280, 3018301, 11134189, 41096528, 151643937, 559640289, 2065192514, 7621289593, 28124714395, 103789150046, 383013144129, 1413437041011, 5216013647648, 19248692843977

Offset: 0

Silvia Heubach (silvi(AT)cine.net), Apr 21 2000

			a(2)=8 as there is one tiling of a 5 X 2 area with only 1 X 1 tiles, 4 tilings with exactly one 2 X 2 tile and 3 tilings with exactly two 2 X 2 tiles.

S. Heubach, Tiling an m-by-n area with squares of size up to k-by-k (m<=5), Congressus Numerantium 140 (1999), 43-64.
R. J. Mathar, Tiling nxm rectangles with 1 X 1 and s X s squares arXiv:1609.03964 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2,7,-2,-3).

Cf. A054854, A000045.

Column k=5 of A245013.

f[{A_, B_}] := Module[{til = A, basic = B}, {Flatten[Append[til, ListConvolve[A, B]]], AppendTo[basic, 2 Fibonacci[Length[B] + 2]]}]; NumOfTilings[n_] := Nest[f, {{1, 1}, {1, 7}}, n - 2][[1]] NumOfTilings[30]

a(n) = b(1)a(n-1)+b(2)a(n-2)+...+b(n)a(0), where a(0)=a(1)=1 and b(1)=1, b(2)=7, b(n)=F(n+1)of A000045 (Fibonacci numbers) for n>2.
a(n) = 2*a(n-1) + 7*a(n-2) - 2*a(n-3) - 3*a(n-4). - Keith Schneider (kschneid(AT)bulldog.unca.edu), Apr 02 2006
G.f.: (1-x-x^2)/(1-2*x-7*x^2+2*x^3+3*x^4). [R. J. Mathar, Nov 02 2008]

A063650 Number of ways to tile a 6 X n rectangle with 1 X 1 and 2 X 2 tiles.

Original entry on oeis.org

1, 1, 13, 43, 269, 1213, 6427, 31387, 159651, 795611, 4005785, 20064827, 100764343, 505375405, 2536323145, 12724855013, 63851706457, 320373303983, 1607526474153, 8065864257905, 40471399479495, 203068825478591, 1018918472214687, 5112520236292975, 25652573037707685

Offset: 0

Reiner Martin, Jul 23 2001

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. J. Mathar, Tiling nXm rectangles with 1X1 and sXs squares arXiv:1609.03964 [math.CO] (2016) Section 4.1.
Index entries for linear recurrences with constant coefficients, signature (2,16,1,-27,1,4).

Cf. A001045, A054854, A054855, A063650-A063654.

Column k=6 of A245013.

I:=[1,1,13,43,269,1213]; [n le 6 select I[n] else 2*Self(n-1)+16*Self(n-2)+Self(n-3)-27*Self(n-4)+Self(n-5)+4*Self(n-6): n in [1..30]]; // Vincenzo Librandi, Oct 30 2018

LinearRecurrence[{2, 16, 1, -27, 1, 4}, {1, 1, 13, 43, 269, 1213}, 22] (* Jean-François Alcover, Oct 30 2018 *)
CoefficientList[Series[(-1+x+5*x^2-x^4)/(-1+2*x+16*x^2+x^3-27*x^4+x^5+4*x^6), {x, 0, 50}], x] (* Stefano Spezia, Oct 30 2018 *)

G.f.: ( -1+x+5*x^2-x^4 ) / ( -1+2*x+16*x^2+x^3-27*x^4+x^5+4*x^6 ).

a(n) = 2a(n-1) + 16a(n-2) + a(n-3) - 27a(n-4) + a(n-5) + 4a(n-6) - Keith Schneider (kschneid(AT)bulldog.unca.edu), Apr 02 2006

A165791 Number of tilings of a 4 X n rectangle using dominoes and right trominoes.

Original entry on oeis.org

1, 1, 11, 55, 380, 2319, 15171, 96139, 619773, 3962734, 25445515, 163048957, 1045897075, 6705473761, 43001795070, 275730928993, 1768128097215, 11337760387473, 72702310606249, 466192677008538, 2989403530821497, 19169143325987983, 122919655766448729

Offset: 0

Alois P. Heinz, Sep 26 2009

			a(2) = 11, because there are 11 tilings of a 4 X 2 rectangle using dominoes and right trominoes:
  .___. .___. .___. ._._. ._._. .___. .___. .___. .___. .___. .___.
  |___| |___| |_._| | | | | | | |___| |___| | ._| |_. | | ._| |_. |
  |___| |_._| | | | |_|_| |_|_| | ._| |_. | |_| | | |_| |_| | | |_|
  |___| | | | |_|_| |___| | | | |_| | | |_| |___| |___| | |_| |_| |
  |___| |_|_| |___| |___| |_|_| |___| |___| |___| |___| |___| |___|  .

Alois P. Heinz, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (4, 21, -25, -65, -17, 24, -11, -15, 9).

Cf. A165716, A127870, A054854, A226322.

Column k=4 of A219987.

a:= n-> (Matrix([[619773, 96139, 15171, 2319, 380, 55, 11, 1, 1]]). Matrix(9, (i,j)-> if i=j-1 then 1 elif j=1 then [4, 21, -25, -65, -17, 24, -11, -15, 9][i] else 0 fi)^n)[1,9]: seq(a(n), n=0..25);

a[n_] := {619773, 96139, 15171, 2319, 380, 55, 11, 1, 1} . MatrixPower[ Table[ Which[i == j-1, 1, j == 1, {4, 21, -25, -65, -17, 24, -11, -15, 9}[[i]], True, 0], {i, 1, 9}, {j, 1, 9}], n] // Last; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 04 2013, translated and adapted from Alois P. Heinz's Maple program *)

G.f.: (2*x^8-5*x^7+2*x^6-x^5-19*x^4-15*x^3+14*x^2+3*x-1) / (9*x^9-15*x^8-11*x^7+24*x^6-17*x^5-65*x^4-25*x^3+21*x^2+4*x-1).

A063654 Number of ways to tile a 10 X n rectangle with 1 X 1 and 2 X 2 tiles.

Original entry on oeis.org

1, 1, 89, 683, 16717, 221799, 4005785, 61643709, 1029574631, 16484061769, 269718819131, 4364059061933, 71019435701025, 1152314664726905, 18725412666911121, 304052089851133193, 4939032362569285343

Offset: 0

Reiner Martin, Jul 23 2001

nonn

Index entries for linear recurrences with constant coefficients, signature (11, 195, -1459, -10427, 79002, 188873, -1922453, -522744, 21726132, -11085988, -137059276, 114023550, 533938164, -503739499, -1345858084, 1272444796, 2223076291, -1997887777, -2382027317, 2015253425, 1613022647, -1309632545, -660948344, 533727589, 152685069, -128889883, -17772195, 16954690, 883980, -1089264, -11392, 26112).

Cf. A001045, A054854, A054855, A063650-A063653.

Column k=10 of A245013.

a(n) = 11a(n-1) + 195a(n-2) - 1459a(n-3) - 10427a(n-4) + 79002a(n-5) + 188873a(n-6) - 1922453a(n-7) - 522744a(n-8) + 21726132a(n-9) - 11085988a(n-10) - 137059276a(n-11) + 114023550a(n-12) + 533938164a(n-13) - 503739499a(n-14) - 1345858084a(n-15) + 1272444796a(n-16) + 2223076291a(n-17) - 1997887777a(n-18) - 2382027317a(n-19) + 2015253425a(n-20) + 1613022647a(n-21) - 1309632545a(n-22) - 660948344a(n-23) + 533727589a(n-24) + 152685069a(n-25) - 128889883a(n-26) - 17772195a(n-27) + 16954690a(n-28) + 883980a(n-29) - 1089264a(n-30) - 11392a(n-31) + 26112a(n-32) - Keith Schneider (kschneid(AT)bulldog.unca.edu), Apr 02 2006

A165799 Number of tilings of a 4 X n rectangle using right trominoes and 2 X 2 tiles.

Original entry on oeis.org

1, 0, 1, 4, 6, 16, 37, 92, 245, 560, 1426, 3720, 9069, 22808, 58177, 145660, 366318, 925536, 2331269, 5872212, 14802941, 37311528, 94038250, 236999064, 597348237, 1505640016, 3794761257, 9564393972, 24106951622, 60759989040, 153141435269, 385986293964

Offset: 0

Alois P. Heinz, Sep 27 2009

			a(4) = 6, because there are 6 tilings of a 4 X 4 rectangle using right trominoes and 2 X 2 tiles:
  .___.___. .___.___. .___.___. .___.___. .___.___. .___.___.
  | . | . | | ._|_. | | ._| . | | ._|_. | | ._|_. | | . |_. |
  |___|___| |_| . |_| |_| |___| |_| ._|_| |_|_. |_| |___| |_|
  | . | . | | |___| | | |___| | | |_| . | | . |_| | | |___| |
  |___|___| |___|___| |___|___| |___|___| |___|___| |___|___|

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,9,1,-3,-22,-16,0,-4).

Cf. A165791, A165716, A054854, A054856, A226322.

Column k=4 of A219946.

a:= n-> (Matrix([[4, 1, 0, 1, 0$5]]). Matrix(9, (i,j)-> if i=j-1 then 1 elif j=1 then [1, 1, 9, 1, -3, -22, -16, 0, -4][i] else 0 fi)^n)[1,4]: seq(a(n), n=0..30);

Series[ (-6*x^3 - x + 1) / (4*x^9 + 16*x^7 + 22*x^6 + 3*x^5 - x^4 - 9*x^3 - x^2 - x + 1), {x, 0, 31}] // CoefficientList[#, x] & (* Jean-François Alcover, Jun 18 2013, after Alois P. Heinz *)
LinearRecurrence[{1,1,9,1,-3,-22,-16,0,-4},{1,0,1,4,6,16,37,92,245},40] (* Harvey P. Dale, Nov 09 2024 *)

G.f.: -(6*x^3+x-1) / (4*x^9+16*x^7+22*x^6+3*x^5-x^4-9*x^3-x^2-x+1).

a(n) = a(n-1) +a(n-2) +9*a(n-3) +a(n-4) -3*a(n-5) -22*a(n-6) -16*a(n-7) -4*a(n-9).

A063652 Number of ways to tile an 8 X n rectangle with 1 X 1 and 2 X 2 tiles.

Original entry on oeis.org

1, 1, 34, 171, 2115, 16334, 159651, 1382259, 12727570, 113555791, 1029574631, 9258357134, 83605623809, 753361554685, 6795928721858, 61270295494859, 552555688390363, 4982395765808506, 44929655655496287, 405145692220245539, 3653405881837027898

Offset: 0

Reiner Martin, Jul 23 2001

nonn

S. Butler and S. Osborne, Counting tilings by taking walks, 2012. - From _N. J. A. Sloane_, Dec 27 2012
R. J. Mathar, Tiling nXm rectangles with 1X1 and sXs squares, arXiv:1609.03964 (2016) Section 4.1
Index entries for linear recurrences with constant coefficients, signature (6, 50, -171, -514, 1800, 845, -5430, 704, 6175, -1762, -2810, 870, 392, -120).

Cf. A001045, A054854, A054855, A063650-A063654.

Column k=8 of A245013.

a(n) = 6a(n-1) + 50a(n-2) - 171a(n-3) - 514a(n-4) + 1800a(n-5) + 845a(n-6) - 5430a(n-7) + 704a(n-8) + 6175a(n-9) - 1762a(n-10) - 2810a(n-11) + 870a(n-12) + 392a(n-13) - 120a(n-14). - Keith Schneider (kschneid(AT)bulldog.unca.edu), Apr 02 2006

G.f.: ( 1 -5*x -22*x^2 +88*x^3 +74*x^4 -378*x^5 -31*x^6 +597*x^7 -114*x^8 -336*x^9 +94*x^10 +52*x^11 -16*x^12 ) / ( 1 -6*x -50*x^2 +171*x^3 +514*x^4 -1800*x^5 -845*x^6 +5430*x^7 -704*x^8 -6175*x^9 +1762*x^10 +2810*x^11 -870*x^12 -392*x^13 +120*x^14 ). - R. J. Mathar, Dec 19 2015

A063653 Number of ways to tile a 9 X n rectangle with 1 X 1 and 2 X 2 tiles.

Original entry on oeis.org

1, 1, 55, 341, 5933, 59925, 795611, 9167119, 113555791, 1355115601, 16484061769, 198549329897, 2403674442213, 29023432116879, 350917980468767, 4239961392742933, 51247532773412135, 619304595300705203, 7484739788762129061, 90454037365096154821

Offset: 0

Reiner Martin, Jul 23 2001

nonn

a(8) is number of ways can kings be placed on an 8 X 8 chessboard so that no two kings can attack each other. - Vaclav Kotesovec, Apr 02 2010

Index entries for linear recurrences with constant coefficients, signature (6, 110, -262, -2786, 5916, 25168, -53907, -95299, 197820, 193866, -340168, -228528, 279636, 137864, -108909, -33736, 20214, 2460, -1296).

Cf. A001045, A054854, A054855, A063650, A063651, A063652, A063654.

Column k=9 of A245013.

a(n) = 6*a(n-1) + 110*a(n-2) - 262*a(n-3) - 2786*a(n-4) + 5916*a(n-5) + 25168*a(n-6) - 53907*a(n-7) - 95299*a(n-8) + 197820*a(n-9) + 193866*a(n-10) - 340168*a(n-11) - 228528*a(n-12) + 279636*a(n-13) + 137864*a(n-14) - 108909*a(n-15) - 33736*a(n-16) + 20214*a(n-17) + 2460*a(n-18) - 1296*a(n-19).

Subscripts in formula repaired by Ron Hardin, Dec 29 2010

cp's OEIS Frontend

A063443 Number of ways to tile an n X n square with 1 X 1 and 2 X 2 tiles.

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A245013 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and 2 X 2 squares; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

Mathematica

A226322 Number of tilings of a 4 X n rectangle using L tetrominoes and 2 X 2 tiles.

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Maple

Mathematica

Formula

A054855 Number of ways to tile a 5 X n area with 1 X 1 and 2 X 2 tiles.

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Mathematica

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A063650 Number of ways to tile a 6 X n rectangle with 1 X 1 and 2 X 2 tiles.

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Magma

Mathematica

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A165791 Number of tilings of a 4 X n rectangle using dominoes and right trominoes.

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Maple

Mathematica

Formula

A063654 Number of ways to tile a 10 X n rectangle with 1 X 1 and 2 X 2 tiles.

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Formula

A165799 Number of tilings of a 4 X n rectangle using right trominoes and 2 X 2 tiles.

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