A219866 -id:A219866 - cp's OEIS frontend

A219987 Number A(n,k) of tilings of a k X n rectangle using dominoes and right trominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 5, 5, 1, 1, 1, 0, 11, 8, 11, 0, 1, 1, 1, 24, 55, 55, 24, 1, 1, 1, 0, 53, 140, 380, 140, 53, 0, 1, 1, 1, 117, 633, 2319, 2319, 633, 117, 1, 1, 1, 0, 258, 1984, 15171, 21272, 15171, 1984, 258, 0, 1

Offset: 0

Alois P. Heinz, Dec 02 2012

			A(3,3) = 8, because there are 8 tilings of a 3 X 3 rectangle using dominoes and right trominoes:
  .___._.   .___._.   .___._.   .___._.
  |___| |   |___| |   |___| |   |_. | |
  | ._|_|   | | |_|   | |___|   | |_|_|
  |_|___|   |_|___|   |_|___|   |_|___|
  ._.___.   ._.___.   ._.___.   ._.___.
  | |___|   | | ._|   | |___|   | |___|
  |___| |   |_|_| |   |_|_. |   |_| | |
  |___|_|   |___|_|   |___|_|   |___|_|
Square array A(n,k) begins:
  1,  1,   1,    1,     1,       1,         1,          1, ...
  1,  0,   1,    0,     1,       0,         1,          0, ...
  1,  1,   2,    5,    11,      24,        53,        117, ...
  1,  0,   5,    8,    55,     140,       633,       1984, ...
  1,  1,  11,   55,   380,    2319,     15171,      96139, ...
  1,  0,  24,  140,  2319,   21272,    262191,    2746048, ...
  1,  1,  53,  633, 15171,  262191,   5350806,  100578811, ...
  1,  0, 117, 1984, 96139, 2746048, 100578811, 3238675344, ...

Liang Kai, Antidiagonals n = 0..35, flattened (Antidiagonals n = 0..31 from Alois P. Heinz)
Kai Liang, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025. See p. 25, Table 4.
Wikipedia, Domino
Wikipedia, Tromino

Columns (or rows) k=0-10 give: A000012, A059841, A052980, A165716, A165791, A219988, A219989, A219990, A219991, A219992, A219993.
Main diagonal gives: A219994.
Cf. A219866, A364457.

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

b[n_, l_] := b[n, l] = Module[{k, t}, If[Max[l] > n, 0, If[n == 0 || l == {}, 1, If[Min[l] > 0, t = Min[l]; b[n-t, l-t], For[k = 1, True, k++, If[l[[k]] == 0, Break[]]]; b[n, ReplacePart[l, k -> 2]] + If[k > 1 && l[[k-1]] == 1, b[n, ReplacePart[l, {k -> 2, k-1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 1, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]] + b[n, ReplacePart[l, {k -> 1, k+1 -> 2}]] + b[n, ReplacePart[l, {k -> 2, k+1 -> 1}]], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2, k+2 -> 2}]] + b[n, ReplacePart[l, {k -> 2, k+1 -> 2, k+2 -> 1}]], 0]]]]]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 05 2013, translated from Alois P. Heinz's Maple program *)

from sage.combinat.tiling import TilingSolver, Polyomino
def A(n,k):
    p = Polyomino([(0,0), (0,1)])
    q = Polyomino([(0,0), (0,1), (1,0)])
    T = TilingSolver([p,q], box=[n,k], reusable=True, reflection=True)
    return T.number_of_solutions()
# Ralf Stephan, May 21 2014

A364457 Number A(n,k) of tilings of a k X n rectangle using dominoes and 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, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 6, 1, 1, 1, 2, 17, 30, 17, 2, 1, 1, 2, 43, 145, 145, 43, 2, 1, 1, 3, 108, 733, 1352, 733, 108, 3, 1, 1, 4, 280, 3540, 12688, 12688, 3540, 280, 4, 1, 1, 5, 727, 17300, 115958, 226922, 115958, 17300, 727, 5, 1

Offset: 0

Alois P. Heinz, Jul 25 2023

			A(3,2) = A(2,3) = 6:
  .___.   .___.   .___.   .___.   .___.   .___.
  | | |   |___|   | | |   |___|   | ._|   |_. |
  | | |   |___|   |_|_|   | | |   |_| |   | |_|
  |_|_|   |___|   |___|   |_|_|   |___|   |___|  .
.
Square array A(n,k) begins:
  1, 1,   1,     1,       1,        1,          1,            1, ...
  1, 0,   1,     1,       1,        2,          2,            3, ...
  1, 1,   2,     6,      17,       43,        108,          280, ...
  1, 1,   6,    30,     145,      733,       3540,        17300, ...
  1, 1,  17,   145,    1352,    12688,     115958,      1075397, ...
  1, 2,  43,   733,   12688,   226922,    3927233,     68846551, ...
  1, 2, 108,  3540,  115958,  3927233,  128441094,   4263997124, ...
  1, 3, 280, 17300, 1075397, 68846551, 4263997124, 267855152858, ...

Alois P. Heinz, Table of n, a(n) for n = 0..350
Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
Wikipedia, Domino (mathematics)
Wikipedia, Tromino

Columns (or rows) k=0-10 give: A000012, A182097(n) = A000931(n+3), A019439, A364460, A364155, A364556, A364616, A364617, A364632, A364638, A364640.
Main diagonal gives A364504.
Cf. A219866, A219987, A233320.

A(n,k) = A(k,n).

Terms n,k>=4 had to be corrected as was pointed out by Martin Fuller and David Radcliffe - Alois P. Heinz, Apr 05 2025

A129682 Number of ways tiling a 2 X n rectangle with 2 X 1 (domino) and 3 X 1 (tromino) tiles.

Original entry on oeis.org

1, 1, 2, 4, 7, 15, 30, 60, 123, 249, 506, 1030, 2093, 4257, 8658, 17606, 35807, 72821, 148096, 301188, 612531, 1245717, 2533444, 5152318, 10478383, 21310119, 43338854, 88139182, 179250591, 364545863, 741384936, 1507770834, 3066386677, 6236177973, 12682652180

Offset: 0

Terry Petrard (temper3243(AT)gmail.com), May 04 2008

Computed using a program with backtracking.

Robert Gerbicz and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..50 from Robert Gerbicz)
Terry Petrard, C program
Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2,1,-1).

Column k=2 of A219866. - Alois P. Heinz, Nov 30 2012

LinearRecurrence[{2,0,1,-2,1,-1},{1,2,4,7,15,30},40] (* Harvey P. Dale, Sep 02 2012 *)

my(a=vector(50)); a[1]=1; a[2]=1;a[3]=2; a[4]=4; a[5]=7; a[6]=15; for(n=7, 50, a[n]=2*a[n-1]+a[n-3]-2*a[n-4]+a[n-5]-a[n-6]); a \\ Robert Gerbicz, May 09 2008

a(n) = a(n-1) + a(n-2) + a(n-3) + 2*r(n-3), where r(n) = r(n-1) + r(n-2) + a(n-2);
f(n) = f(n-1) + p(n) + q(n), where p(n) is the number of ways after filling 2 X n with a horizontal 2 X 1 domino and q(n) is the number of ways after filling 2 X n with a horizontal 3 X 1 domino.
r(n) is a 2 X n rectangle with 1 square removed from top left
p(n) is a 2 X n rectangle with 2 square removed from top left
q(n) is a 2 X n rectangle with 3 square removed from top left
p(n) = f(n-2) + r(n-2) (tiling with 2x1 gives f(n-2) and 3x1 gives r(n-2))
q(n) = f(n-3) + r(n-2) (tiling with 3x1 gives f(n-3) and 2x1 gives r(n-2))
r(n) = r(n-1) + p(n-2) (tiling with 2x1 gives r(n-1), tiling with a 3x1 gives p(n-2))
a(n)=2*a(n-1)+a(n-3)-2*a(n-4)+a(n-5)-a(n-6) - Robert Gerbicz, May 09 2008
G.f.: (1 - x - x^3)/((1-x)*(1-x-x^2-2*x^3-x^5)). - R. J. Mathar, Oct 30 2008

More terms from Robert Gerbicz, May 09 2008

A219867 Number of tilings of a 3 X n rectangle using dominoes and straight (3 X 1) trominoes.

Original entry on oeis.org

1, 1, 4, 14, 41, 143, 472, 1562, 5233, 17395, 58002, 193346, 644219, 2147421, 7156704, 23852324, 79497767, 264952955, 883057354, 2943113598, 9809007073, 32692164351, 108958689984, 363145140266, 1210315480391, 4033823637937, 13444208923518, 44807796457932

Offset: 0

Alois P. Heinz, Nov 30 2012

			a(2) = 4, because there are 4 tilings of a 3 X 2 rectangle using dominoes and straight (3 X 1) trominoes:
.___.   .___.   .___.   .___.
| | |   |___|   | | |   |___|
| | |   |___|   |_|_|   | | |
|_|_|   |___|   |___|   |_|_|

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,12,-3,-11,-30,-5,-13,24,14,24,-3,3,-10,-5,-4,-1,-1).

Column k=3 of A219866.

gf:= -(x^15 +x^13 +x^12 +6*x^11 -x^10 +3*x^9 -10*x^8 -4*x^7 -9*x^6 +2*x^5 +2*x^4 +7*x^3 +2*x^2 -1) / (x^18 +x^17 +4*x^16 +5*x^15 +10*x^14 -3*x^13 +3*x^12 -24*x^11 -14*x^10 -24*x^9 +13*x^8 +5*x^7 +30*x^6 +11*x^5 +3*x^4 -12*x^3 -5*x^2 -x +1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

G.f.: see Maple program.

A219874 Number of tilings of an n X n square using dominoes and straight (3 X 1) trominoes.

Original entry on oeis.org

1, 0, 2, 14, 184, 9612, 1143834, 354859954, 295743829064, 631206895803116, 3541054185616706122, 51821077154605344550820, 1976225122734369352127065686, 196913655491597719598898811003348, 51179690353659852099434654264900753288, 34716223657627061096793572212632925410608268

Offset: 0

Alois P. Heinz, Nov 30 2012

nonn

			a(3) = 14, because there are 14 tilings of a 3 X 3 square using dominoes and straight (3 X 1) trominoes:
  ._____. ._____. ._____. ._____. .___._. .___._. .___._.
  | | | | | | | | | |___| | |___| | | | | |___| | |___| |
  | | | | | |_|_| | |___| | | | | |_|_| | |___| | | | | |
  |_|_|_| |_|___| |_|___| |_|_|_| |___|_| |___|_| |_|_|_|
  ._____. ._____. ._____. ._____. ._____. ._____. ._____.
  |_____| |_____| |_____| |_____| | |___| | | | | |___| |
  |_____| | |___| | | | | |___| | |_|___| |_|_|_| |___|_|
  |_____| |_|___| |_|_|_| |___|_| |_____| |_____| |_____|  .

Kai Liang, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025. See p. 25, Table 4.

Main diagonal of A219866.

Cf. A233807, A364504.

a(12) from Alois P. Heinz, Sep 30 2014

a(13)-a(15) (using Liang Kai's terms in A219866) from Alois P. Heinz, Mar 12 2025

A233313 Number of tilings of a 2 X 3 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.

Original entry on oeis.org

1, 4, 45, 717, 9787, 148414, 2282036, 34688229, 530613082, 8119995275, 124183342755, 1899899589557, 29066650643742, 444678773140018, 6803102237763707, 104079849391557116, 1592303310404361651, 24360457647669398381, 372687643806340329749, 5701702230014416236396

Offset: 0

Alois P. Heinz, Dec 07 2013

nonn

			a(1) = A219866(3,2) = A129682(3) = A219866(2,3) = A219867(2) = 4:
._____.  ._____.  ._____.  ._____.
|_____|  | | | |  |___| |  | |___|
|_____|  |_|_|_|  |___|_|  |_|___|.

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

Cf. A000931, A129682, A219866, A219867, A233505, A233506, A233507, A233509.

s:= subsop:
b:= proc(n, l) option remember; local k, t; t:= min(l[]);
      if n=0 then 1 elif t>0 then b(n-t, map(h->h-t, l))
    else for k while l[k]>0 do od; add(`if`(n>=j,
         b(n, s(k=j, l)), 0), j=2..3)+ `if`(k<=4 and l[k+2]=0,
         b(n, s(k=1, k+2=1, l))+ `if`(k<=2 and l[k+4]=0,
         b(n, s(k=1, k+2=1, k+4=1, l)), 0), 0)+ `if`(
         irem(k, 2)>0 and l[k+1]=0, b(n, s(k=1, k+1=1, l)), 0)
      fi
    end:
a:=n-> b(n, [0$6]):
seq(a(n), n=0..25);

b[n_, l_] := b[n, l] = Module[{k, t}, t = Min[l]; If [n == 0, 1, If[t > 0, b[n-t, l-t], k = 1; While[l[[k]] > 0 , k++]; Sum[If[n >= j, b[n, ReplacePart[l, k -> j]], 0], {j, 2, 3}] + If[k <= 4 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 1, k+2 -> 1}]] + If[k <= 2 && l[[k+4]] == 0, b[n, ReplacePart[l, {k -> 1, k+2 -> 1, k+4 -> 1}]], 0], 0] + If[Mod[k, 2] > 0 && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 0] ] ] ]; a[n_] := b[n, Array[0&, 6]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A233505 Number of tilings of a 2 X 2 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.

Original entry on oeis.org

1, 2, 9, 45, 201, 1062, 5564, 28859, 152012, 799387, 4202165, 22117465, 116385352, 612443308, 3223118545, 16961953022, 89264218645, 469766599585, 2472212575433, 13010374123502, 68468999197712, 360328057238019, 1896278913278432, 9979444454056631

Offset: 0

Alois P. Heinz, Dec 11 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, 0, 19, -62, 28, -141, 20, -92, 156, 76, 184, 62, 66, 41, 27, -12, 9, 2, -2, -1).

Cf. A000931, A129682, A219866, A219867, A233313, A233506, A233507, A233509.

a:= n-> coeff(series((x^17 +3*x^16 -x^15 -5*x^14 +8*x^13 -8*x^12 -22*x^10 -20*x^9 -62*x^8 -20*x^7 -42*x^6 +18*x^5 +19*x^3 +x^2 +3*x -1)/(-x^20 -2*x^19 +2*x^18 +9*x^17 -12*x^16 +27*x^15 +41*x^14 +66*x^13 +62*x^12 +184*x^11 +76*x^10 +156*x^9 -92*x^8 +20*x^7 -141*x^6 +28*x^5 -62*x^4 +19*x^3 +5*x -1), x, n+1), x, n):
seq(a(n), n=0..40);

G.f.: (x^17 +3*x^16 -x^15 -5*x^14 +8*x^13 -8*x^12 -22*x^10 -20*x^9 -62*x^8 -20*x^7 -42*x^6 +18*x^5 +19*x^3 +x^2 +3*x -1) / (-x^20 -2*x^19 +2*x^18 +9*x^17 -12*x^16 +27*x^15 +41*x^14 +66*x^13 +62*x^12 +184*x^11 +76*x^10 +156*x^9 -92*x^8 +20*x^7 -141*x^6 +28*x^5 -62*x^4 +19*x^3 +5*x -1).

A233506 Number of tilings of a 3 X 3 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.

Original entry on oeis.org

1, 14, 717, 62253, 4732061, 382882762, 31449389548, 2571574546111, 210607584419520, 17254476918858789, 1413637025226131703, 115812392270890399373, 9488271882367228634756, 777357166136453697810804, 63686950935296529029018801, 5217741644362129948411085318

Offset: 0

Alois P. Heinz, Dec 11 2013

nonn

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

Cf. A000931, A129682, A219866, A219867, A233313, A233505, A233507, A233509.

b:= proc(n, l) option remember; local k, t; t:= min(l[]);
      if n=0 then 1 elif t>0 then b(n-t, map(h->h-t, l))
    else for k while l[k]>0 do od;
         add(`if`(n>=j, b(n, s(k=j, l)), 0), j=2..3)+
         `if`(k<=6 and l[k+3]=0, b(n, s(k=1, k+3=1, l)), 0)+
         `if`(k<=3 and l[k+3]=0 and l[k+6]=0,
            b(n, s(k=1, k+3=1, k+6=1, l)), 0)+
         `if`(irem(k, 3)>0 and l[k+1]=0,
            b(n, s(k=1, k+1=1, l)), 0)+
         `if`(irem(k, 3)=1 and l[k+1]=0 and l[k+2]=0,
            b(n, subsop(k=1, k+1=1, k+2=1, l)), 0)
      fi
    end:
a:=n-> b(n, [0$9]): s:=subsop:
seq(a(n), n=0..10);

b[n_, l_] := b[n, l] = Module[{k, t}, t := Min[l]; If [n == 0, 1, If[t > 0, b[n-t, l-t], k = 1; While[l[[k]] > 0, k++ ]; Sum[If[n >= j, b[n, ReplacePart[l, k -> j]], 0], {j, 2, 3}] + If[k <= 6 && l[[k+3]] == 0, b[n, ReplacePart[l, {k -> 1, k+3 -> 1}]], 0] + If[k <= 3 && l[[k+3]] == 0 && l[[k+6]] == 0, b[n, ReplacePart[l, {k -> 1, k+3 -> 1, k+6 -> 1}]], 0] + If[Mod[k, 3] > 0 && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 0] + If[Mod[k, 3] == 1 && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1, k+2 -> 1}]], 0] ]] ]; a[n_] := b[n, Array[0&, 9]]; Table[a[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A233507 Number of tilings of a 2 X 4 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.

Original entry on oeis.org

1, 7, 201, 9787, 379688, 16512483, 726964790, 31549810845, 1378740599284, 60239603421159, 2630166605483293, 114886450998314920, 5017916294582867990, 219163121582772423673, 9572435654283943792842, 418094220600909382190818, 18261053013117932038592765

Offset: 0

Alois P. Heinz, Dec 11 2013

nonn

			a(1) = A219866(4,2) = A129682(4) = A219866(2,4) = A219862(2) = 7:
._______. ._______. ._______. ._______.
|_____| | | |_____| | | | | | |___| | |
|_____|_| |_|_____| |_|_|_|_| |___|_|_|
._______. ._______. ._______.
| |___| | | | |___| |___|___|
|_|___|_| |_|_|___| |___|___|.

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

Cf. A000931, A129682, A219866, A219867, A233313, A233505, A233506, A233509.

b:= proc(n, l) option remember; local k, t; t:= min(l[]);
      if n=0 then 1
    elif t>0 then b(n-t, map(h->h-t, l))
    else for k while l[k]>0 do od;
         add(`if`(n>=j, b(n, s(k=j, l)), 0), j=2..3)+
         `if`(k<=6 and l[k+2]=0, b(n, s(k=1, k+2=1, l)), 0)+
         `if`(k<=4 and l[k+2]=0 and l[k+2*2]=0, b(n, s(k=1,
         k+2=1, k+2*2=1, l)), 0)+ `if`(irem(k, 2)>0 and
         l[k+1]=0, b(n, s(k=1, k+1=1, l)), 0)
      fi
    end:
a:=n-> b(n, [0$8]): s:= subsop:
seq(a(n), n=0..10);

b[n_, l_] := b[n, l] = Module[{k, t}, t = Min[l]; Which[n == 0, 1, t > 0, b[n-t, l-t], True, For[k = 1, l[[k]] > 0, k++]; Sum[If[n >= j, b[n, ReplacePart[l, k -> j]], 0], {j, 2, 3}] + If[k <= 6 && l[[k + 2]] == 0, b[n, ReplacePart[l, {k -> 1, k+2 -> 1}]], 0] + If[k <= 4 && l[[k+2]] == 0 && l[[k+2*2]] == 0, b[n, ReplacePart[l, {k -> 1, k+2 -> 1, k+2*2 -> 1}]], 0] + If[Mod[k, 2] > 0 && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 0]]]; a[n_] := b[n, Array[0&, 8]]; Table[Print[an = a[n]]; an, {n, 0, 16}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)

A233509 Number of tilings of a 2 X 5 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.

Original entry on oeis.org

1, 15, 1062, 148414, 16512483, 2043497465, 257251613508, 31941208907916, 3990164870713039, 498504394558488109, 62237975023439983192, 7773270324407375580946, 970802515607358269506951, 121240108673115249961266051, 15141593230837339625055971170

Offset: 0

Alois P. Heinz, Dec 11 2013

nonn

			a(1) = A219866(5,2) = A129682(5) = A219866(2,5) = A219868(2) = 15:
.___.  .___.  .___.  .___.  .___.  .___.  .___.  .___.
| | |  |___|  | | |  |___|  | | |  |___|  | | |  |___|
| | |  |___|  |_|_|  | | |  | | |  |___|  |_|_|  | | |
|_|_|  |___|  |___|  |_|_|  |_|_|  |___|  |___|  |_|_|
| | |  | | |  | | |  | | |  |___|  |___|  |___|  |___|
|_|_|  |_|_|  |_|_|  |_|_|  |___|  |___|  |___|  |___|
.___.  .___.  .___.  .___.  .___.  .___.  .___.
| | |  | | |  |___|  |___|  | | |  | | |  |___|
|_|_|  |_|_|  |___|  |___|  |_| |  | |_|  | | |
| | |  | | |  | | |  | | |  | |_|  |_| |  | | |
| | |  |_|_|  | | |  |_|_|  | | |  | | |  |_|_|
|_|_|  |___|  |_|_|  |___|  |_|_|  |_|_|  |___|.

Alois P. Heinz, Table of n, a(n) for n = 0..40

Cf. A000931, A129682, A219866, A219867, A233313, A233505, A233506, A233507.

b:= proc(n, l) option remember; local k, t; t:= min(l[]);
      if n=0 then 1 elif t>0 then b(n-t, map(h->h-t, l))
    else for k while l[k]>0 do od;
         add(`if`(n>=j, b(n, s(k=j, l)), 0), j=2..3)+
         `if`(k<=5 and l[k+5]=0, b(n, s(k=1, k+5=1, l)), 0)+
         `if`(irem(k, 5)>0 and l[k+1]=0, b(n, s(k=1, k+1=1, l)), 0)+
         `if`(irem(k, 5) in [$1..3] and l[k+1]=0 and l[k+2]=0,
         b(n, s(k=1, k+1=1, k+2=1, l)), 0)
      fi
    end:
a:=n-> b(n, [0$10]): s:=subsop:
seq(a(n), n=0..4);

b[n_, l_] := b[n, l] = Module[{k, t}, t = Min[l]; Which[n == 0, 1, t > 0, b[n-t, l-t], True, For[k = 1, l[[k]] > 0, k++]; Sum[If[n >= j, b[n, ReplacePart[l, k -> j]], 0], {j, 2, 3}] + If[k <= 5 && l[[k+5]] == 0, b[n, ReplacePart[l, {k -> 1, k+5 -> 1}]], 0] + If[Mod[k, 5] > 0 && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 0] + If[1 <= Mod[k, 5] <= 3 && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1, k+2 -> 1}]], 0]]];a[n_] := b[n, Array[0&, 10]]; Table[Print[an = a[n]]; an, {n, 0, 14}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)

cp's OEIS Frontend

A219987 Number A(n,k) of tilings of a k X n rectangle using dominoes and right trominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A364457 Number A(n,k) of tilings of a k X n rectangle using dominoes and trominoes (of any shape); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A129682 Number of ways tiling a 2 X n rectangle with 2 X 1 (domino) and 3 X 1 (tromino) tiles.

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A219867 Number of tilings of a 3 X n rectangle using dominoes and straight (3 X 1) trominoes.

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A219874 Number of tilings of an n X n square using dominoes and straight (3 X 1) trominoes.

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A233313 Number of tilings of a 2 X 3 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.

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A233505 Number of tilings of a 2 X 2 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.

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A233506 Number of tilings of a 3 X 3 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.

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A233507 Number of tilings of a 2 X 4 X n box using bricks of shape 3 X 1 X 1 and 2 X 1 X 1.