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

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.

The table is read by rows. Reading the sequence {T(k,n)}, n=0,1,2,..., use T(n,k) instead of T(k,n) for n>k.

Links

Table of n, a(n) for n=0..44.

Gerhard Kirchner, Maxima code

Example

Triangle T(k,n) begins
k\n_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

T(1,n) = A000045(n+1), Fibonacci numbers.

T(2,n) = A052543(n).

T(3,n) = A226351(n).

T(4,n) = A352590(n).

T(5,n) = A352591(n).

T(n,n) gives A353777.

Sequence in context: A143198 A100064 A153188 * A275980 A343918 A156029

Adjacent sequences: A352586 A352587 A352588 * A352590 A352591 A352592

Keyword

nonn,tabl

Author

Gerhard Kirchner, Mar 22 2022

Status

approved