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

0,5

Comments

For the tiling algorithm, see A351322.

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

Links

Liang Kai, Rows n=0..27, flattened

Gerhard Kirchner, Maxima code

Example

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

Maple

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`(k<nops(l)
            and l[k+1]=0, b(n, subsop(k=1, k+1=1, l))+
         `if`(n>1, 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

Mathematica

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 *)

Prog

(Maxima) /* See Maxima code link. */

Crossrefs

Row/columns 0..5 are A000012, A000045(n+1), A052543, A226351, A352590, A352591.

Main diagonal is A353777.

Cf. A351322.

Sequence in context: A143198 A100064 A153188 * A275980 A394536 A343918

Adjacent sequences: A352586 A352587 A352588 * A352590 A352591 A352592

Keyword

nonn,tabl

Author

Gerhard Kirchner, Mar 22 2022

Status

approved