A219975 -id:A219975 - 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 *)

A233807 Number of tilings of an n X n square using right trominoes and at most one monomino.

Original entry on oeis.org

1, 1, 4, 0, 16, 128, 162, 34528, 943096, 1193600, 3525377600, 480585761344, 2033502499954, 46983507796973152, 32908187880881958736, 458324092996867592192, 83153202122213272708832688, 299769486068040749617049301344

Offset: 0

Alois P. Heinz, Dec 16 2013

nonn

			a(2) = 4:
  .___.  .___.  .___.  .___.
  |_| |  | |_|  | ._|  |_. |
  |___|  |___|  |_|_|  |_|_| .

Wikipedia, Tromino

Cf. A000105, A219874, A219952, A219975, A219994, A220061.

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

$RecursionLimit = 1000; s = ReplacePart; b[n_, w_, l_] := b[n, w, l] = Module[{k, t}, Which[Max[l] > n, 0,n == 0, 1,Min[l] > 0, t = Min[l]; b[n-t, w, l-t], True, For[k = 1, l[[k]] > 0, k++ ]; If[w, b[n, False, s[l, k -> 1]], 0]+If[k > 1 && l[[k-1]] == 1, b[n, w, s[l, {k -> 2, k-1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 1, b[n, w, s[l, {k -> 2, k+1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 0, b[n, w, s[l, {k -> 1, k+1 -> 2}]]+b[n, w, s[l, {k -> 2, k+1 -> 1}]] + If[w, b[n, False, s[l, {k -> 2, k+1 -> 2}]], 0], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, w, s[l, {k -> 2, k+1 -> 2, k+2 -> 2}]], 0] ] ]; a[n_] := b[n, Mod[n, 3] > 0, Array[0 &, n]]; Table[Print[an = a[n]]; an, {n, 0, 16}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)

a(17) from Alois P. Heinz, Sep 24 2014

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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