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%I #24 May 16 2022 09:33:29
%S 1,1,1,1,2,8,1,3,26,163,1,5,90,1125,15623,1,8,306,7546,210690,5684228,
%T 1,13,1046,51055,2865581,154869092,8459468955,1,21,3570,344525,
%U 38879777,4207660108,460706560545,50280716999785,1,34,12190,2326760,527889422,114411435032,25111681648122,5492577770367562,1202536689448371122
%N 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.
%C For the tiling algorithm, see A351322.
%C 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.
%H Gerhard Kirchner, <a href="/A352589/a352589_1.txt">Maxima code</a>
%e Triangle T(k,n) begins
%e k\n_0__1____2______3________4__________5____________6
%e 0: 1
%e 1: 1 1
%e 2: 1 2 8
%e 3: 1 3 26 163
%e 4: 1 5 90 1125 15623
%e 5: 1 8 306 7546 210690 5684228
%e 6: 1 13 1046 51055 2865581 154869092 8459468955
%p b:= proc(n, l) option remember; local k, t;
%p if n=0 or l=[] then 1
%p elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
%p else for k while l[k]>0 do od; b(n, subsop(k=1, l))+
%p `if`(n>1, b(n, subsop(k=2, l)), 0)+ `if`(k<nops(l)
%p and l[k+1]=0, b(n, subsop(k=1, k+1=1, l))+
%p `if`(n>1, b(n, subsop(k=2, k+1=2, l)), 0), 0)
%p fi
%p end:
%p T:= (n, k)-> b(max(n, k), [0$min(n, k)]):
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, May 06 2022
%t b[n_, l_List] := b[n, l] = Module[{k, t}, Which[
%t n == 0 || l == {}, 1,
%t Min[l] > 0, t = Min[l]; b[n - t, l - t],
%t True, For[k = 1, l[[k]] > 0, k++]; b[n, ReplacePart[l, k -> 1]] +
%t If[n > 1, b[n, ReplacePart[l, k -> 2]], 0] + If[k < Length[l] &&
%t l[[k + 1]] == 0, b[n, ReplacePart[l, {k -> 1, k + 1 -> 1}]] +
%t If[n > 1, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0], 0]]];
%t T[n_, k_] := b[Max[n, k], Array[0&, Min[n, k]]];
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, May 16 2022, after _Alois P. Heinz_ *)
%o (Maxima) See Maxima code link.
%Y T(1,n) = A000045(n+1), Fibonacci numbers.
%Y T(2,n) = A052543(n).
%Y T(3,n) = A226351(n).
%Y T(4,n) = A352590(n).
%Y T(5,n) = A352591(n).
%Y T(n,n) gives A353777.
%K nonn,tabl
%O 0,5
%A _Gerhard Kirchner_, Mar 22 2022
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