A202536 -id:A202536 - cp's OEIS frontend

A219967 Number A(n,k) of tilings of a k X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 0, 2, 4, 3, 4, 2, 0, 1, 1, 0, 3, 8, 8, 8, 8, 3, 0, 1, 1, 1, 4, 13, 21, 28, 21, 13, 4, 1, 1, 1, 0, 5, 19, 31, 65, 65, 31, 19, 5, 0, 1, 1, 0, 7, 35, 70, 170, 267, 170, 70, 35, 7, 0, 1

Offset: 0

Alois P. Heinz, Dec 02 2012

			A(4,4) = 3, because there are 3 tilings of a 4 X 4 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:
  ._._____.  ._____._.  ._._._._.
  | |_____|  |_____| |  | . | . |
  | | . | |  | | . | |  |___|___|
  |_|___| |  | |___|_|  | . | . |
  |_____|_|  |_|_____|  |___|___| .
Square array A(n,k) begins:
  1,  1,  1,  1,  1,   1,    1,     1,     1, ...
  1,  0,  0,  1,  0,   0,    1,     0,     0, ...
  1,  0,  1,  1,  1,   2,    2,     3,     4, ...
  1,  1,  1,  2,  3,   4,    8,    13,    19, ...
  1,  0,  1,  3,  3,   8,   21,    31,    70, ...
  1,  0,  2,  4,  8,  28,   65,   170,   456, ...
  1,  1,  2,  8, 21,  65,  267,   804,  2530, ...
  1,  0,  3, 13, 31, 170,  804,  2744, 12343, ...
  1,  0,  4, 19, 70, 456, 2530, 12343, 66653, ...

Liang Kai, Antidiagonals n = 0..35, flattened (Antidiagonals 0..27 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, Tromino

Columns (or rows) k=0-10 give: A000012, A079978, A000931(n+3), A219968, A202536, A219969, A219970, A219971, A219972, A219973, A219974.

Main diagonal gives: A219975.

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=3, l))+
         `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], k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 3]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1, 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 16 2013, translated from Maple *)

A219862 Number of tilings of a 4 X n rectangle using dominoes and straight (3 X 1) trominoes.

Original entry on oeis.org

1, 1, 7, 41, 184, 1069, 5624, 29907, 161800, 862953, 4631107, 24832532, 133028028, 713283085, 3822965706, 20491221900, 109840081931, 588746006676, 3155783700063, 16915482096570, 90669231898345, 486001022349368, 2605035346917456, 13963368769216664

Offset: 0

Alois P. Heinz, Nov 29 2012

			a(2) = 7, because there are 7 tilings of a 4 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 (4, 6, 27, -102, 17, -395, 797, -644, 2838, -2657, 2523, -8602, 4310, -8873, 16066, -2769, 18628, -15477, 3863, -27627, 5040, -9300, 19846, 1875, 15731, -6435, 1924, -7786, -680, -4783, 1842, -657, 1108, 32, 734, -278, 88, -32, 10, -58, 13, -3, -1, 0, 1).

Cf. A165716, A165791, A165799, A190759, A202536.

Column k=4 of A219866.

gf:= -(x^42 +x^41 -4*x^40 +4*x^38 -41*x^37 +16*x^36 +45*x^35 +67*x^34 -166*x^33 +282*x^32 -148*x^31 +155*x^30 -405*x^29 +995*x^28 -1118*x^27 +575*x^26 -1863*x^25 +402*x^24 -3552*x^23 +2577*x^22 -406*x^21 +5797*x^20 -741*x^19 +3045*x^18 -5606*x^17 +223*x^16 -4294*x^15 +2924*x^14 -753*x^13 +3011*x^12 -1029*x^11 +811*x^10 -1205*x^9 +248*x^8 -310*x^7 +229*x^6 -17*x^5 +53*x^4 -20*x^3 -3*x^2 -3*x +1) /
(x^45 -x^43 -3*x^42 +13*x^41 -58*x^40 +10*x^39 -32*x^38 +88*x^37 -278*x^36 +734*x^35 +32*x^34 +1108*x^33 -657*x^32 +1842*x^31 -4783*x^30 -680*x^29 -7786*x^28 +1924*x^27 -6435*x^26 +15731*x^25 +1875*x^24 +19846*x^23 -9300*x^22 +5040*x^21 -27627*x^20 +3863*x^19 -15477*x^18 +18628*x^17 -2769*x^16 +16066*x^15 -8873*x^14 +4310*x^13 -8602*x^12 +2523*x^11 -2657*x^10 +2838*x^9 -644*x^8 +797*x^7 -395*x^6 +17*x^5 -102*x^4 +27*x^3 +6*x^2 +4*x -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..30);

G.f.: see Maple program.

cp's OEIS Frontend

A219967 Number A(n,k) of tilings of a k X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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