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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.
16
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
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
KEYWORD
nonn,tabl
AUTHOR
Gerhard Kirchner, Mar 22 2022
STATUS
approved