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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.

%I A352589 #41 Apr 05 2025 23:12:11
%S A352589 1,1,1,1,2,8,1,3,26,163,1,5,90,1125,15623,1,8,306,7546,210690,5684228,
%T A352589 1,13,1046,51055,2865581,154869092,8459468955,1,21,3570,344525,
%U A352589 38879777,4207660108,460706560545,50280716999785,1,34,12190,2326760,527889422,114411435032,25111681648122,5492577770367562,1202536689448371122
%N 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.
%C A352589 For the tiling algorithm, see A351322.
%C A352589 Reading the sequence {T(n,k)} for k>n, use T(k,n) instead of T(n,k).
%H A352589 Liang Kai, <a href="/A352589/b352589.txt">Rows n=0..27, flattened</a>
%H A352589 Gerhard Kirchner, <a href="/A352589/a352589_1.txt">Maxima code</a>
%e A352589 Triangle T(n,k) begins
%e A352589   n\k_0__1____2______3________4__________5___________6
%e A352589   0:  1
%e A352589   1:  1  1
%e A352589   2:  1  2    8
%e A352589   3:  1  3   26    163
%e A352589   4:  1  5   90   1125    15623
%e A352589   5:  1  8  306   7546   210690    5684228
%e A352589   6:  1 13 1046  51055  2865581  154869092  8459468955
%p A352589 b:= proc(n, l) option remember; local k, t;
%p A352589       if n=0 or l=[] then 1
%p A352589     elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
%p A352589     else for k while l[k]>0 do od; b(n, subsop(k=1, l))+
%p A352589          `if`(n>1, b(n, subsop(k=2, l)), 0)+ `if`(k<nops(l)
%p A352589             and l[k+1]=0, b(n, subsop(k=1, k+1=1, l))+
%p A352589          `if`(n>1, b(n, subsop(k=2, k+1=2, l)), 0), 0)
%p A352589       fi
%p A352589     end:
%p A352589 T:= (n, k)-> b(max(n, k), [0$min(n, k)]):
%p A352589 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, May 06 2022
%t A352589 b[n_, l_List] := b[n, l] = Module[{k, t}, Which[
%t A352589      n == 0 || l == {}, 1,
%t A352589      Min[l] > 0, t = Min[l]; b[n - t, l - t],
%t A352589      True, For[k = 1, l[[k]] > 0, k++]; b[n, ReplacePart[l, k -> 1]] +
%t A352589            If[n > 1, b[n, ReplacePart[l, k -> 2]], 0] + If[k < Length[l] &&
%t A352589            l[[k + 1]] == 0, b[n, ReplacePart[l, {k -> 1, k + 1 -> 1}]] +
%t A352589            If[n > 1, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0], 0]]];
%t A352589 T[n_, k_] := b[Max[n, k], Array[0&, Min[n, k]]];
%t A352589 Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, May 16 2022, after _Alois P. Heinz_ *)
%o A352589 (Maxima) /* See Maxima code link. */
%Y A352589 Row/columns 0..5 are A000012, A000045(n+1), A052543, A226351, A352590, A352591.
%Y A352589 Main diagonal is A353777.
%Y A352589 Cf. A351322.
%K A352589 nonn,tabl
%O A352589 0,5
%A A352589 _Gerhard Kirchner_, Mar 22 2022