A219968 -id:A219968 - 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 *)

A376031 Number of ways to tile a 3 x (2*n) rectangle with dominoes and T's.

Original entry on oeis.org

1, 3, 18, 112, 692, 4294, 26624, 165086, 1023662, 6347440, 39358774, 244053158, 1513307844, 9383614226, 58185263358, 360791140032, 2237168644134, 13872079956206, 86017029971684, 533368425534858, 3307273890427894, 20507514248408832, 127161570097317790

Offset: 0

Greg Dresden and Lucas MingQu Xia, Sep 06 2024

a(n) is the number of ways to tile a 3 X (2*n) rectangle with two kinds of tiles: dominoes (made up of 2 cells) and T's (made up of 4 cells), each of which can be rotated as needed.

			For n=3, here is one of the a(3) = 112 ways to tile a 3 x 6 rectangle using our dominoes and T's:
 ___________
| |___| | | |
|  _|_  |_|_|
|_|___|_|___|.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,7,4,-8,2).

Cf. A033506, A219968, A337492, A346054.

LinearRecurrence[{5, 7, 4, -8, 2}, {1, 3, 18, 112, 692}, 30] (* Paolo Xausa, Sep 23 2024 *)

a(n) = 5*a(n-1) + 7*a(n-2) + 4*a(n-3) - 8*a(n-4) + 2*a(n-5).

G.f.: -(2*x^4-3*x^3-4*x^2-2*x+1)/(2*x^5-8*x^4+4*x^3+7*x^2+5*x-1).

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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