A352589 - OEIS

%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