A353777 -id:A353777 - cp's OEIS frontend

A352589 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using 2 X 2 and 1 X 1 tiles and dominoes, n >= 0, k = 0..n.

1, 1, 1, 1, 2, 8, 1, 3, 26, 163, 1, 5, 90, 1125, 15623, 1, 8, 306, 7546, 210690, 5684228, 1, 13, 1046, 51055, 2865581, 154869092, 8459468955, 1, 21, 3570, 344525, 38879777, 4207660108, 460706560545, 50280716999785, 1, 34, 12190, 2326760, 527889422, 114411435032, 25111681648122, 5492577770367562, 1202536689448371122

Offset: 0

Gerhard Kirchner, Mar 22 2022

For the tiling algorithm, see A351322.

Reading the sequence {T(n,k)} for k>n, use T(k,n) instead of T(n,k).

			Triangle T(n,k) begins
  n\k_0__1____2______3________4__________5___________6
  0:  1
  1:  1  1
  2:  1  2    8
  3:  1  3   26    163
  4:  1  5   90   1125    15623
  5:  1  8  306   7546   210690    5684228
  6:  1 13 1046  51055  2865581  154869092  8459468955

Liang Kai, Rows n=0..27, flattened
Gerhard Kirchner, Maxima code

Row/columns 0..5 are A000012, A000045(n+1), A052543, A226351, A352590, A352591.
Main diagonal is A353777.
Cf. A351322.

b:= proc(n, l) option remember; local k, t;
      if 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 while l[k]>0 do od; b(n, subsop(k=1, l))+
         `if`(n>1, b(n, subsop(k=2, l)), 0)+ `if`(k1, b(n, subsop(k=2, k+1=2, l)), 0), 0)
      fi
    end:
T:= (n, k)-> b(max(n, k), [0$min(n, k)]):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, May 06 2022

b[n_, l_List] := b[n, l] = Module[{k, t}, Which[
     n == 0 || l == {}, 1,
     Min[l] > 0, t = Min[l]; b[n - t, l - t],
     True, For[k = 1, l[[k]] > 0, k++]; b[n, ReplacePart[l, k -> 1]] +
           If[n > 1, b[n, ReplacePart[l, k -> 2]], 0] + If[k < Length[l] &&
           l[[k + 1]] == 0, b[n, ReplacePart[l, {k -> 1, k + 1 -> 1}]] +
           If[n > 1, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0], 0]]];
T[n_, k_] := b[Max[n, k], Array[0&, Min[n, k]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

Maxima
```
/* See Maxima code link. */
```

A353934 Number of tilings of an n X n square using right trominoes, dominoes, and monominoes.

Original entry on oeis.org

1, 1, 11, 369, 83374, 90916452, 546063639624, 17259079054003609, 2916019543694306398589, 2620143594924539083433405392, 12541344781693990981151732534871036, 319608708168951734031266758322647453517098, 43373075269161087186367095378869660507262626652634

Offset: 0

Alois P. Heinz, May 11 2022

nonn

			a(2) = 11:
  .___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
  |_|_| |___| | | | |_|_| |___| |_| | | |_| |_| | |_. | | ._| | |_|
  |_|_| |___| |_|_| |___| |_|_| |_|_| |_|_| |___| |_|_| |_|_| |___| .

Alois P. Heinz, Table of n, a(n) for n = 0..16
Wikipedia, Polyomino

Main diagonal of A353877.

Cf. A004003, A028420, A063443, A219874, A219952, A219994, A220061, A220778, A233807, A270071, A353777.

a(n) = A353877(n,n).

cp's OEIS Frontend

A352589 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using 2 X 2 and 1 X 1 tiles and dominoes, n >= 0, k = 0..n.

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