A226545 Number A(n,k) of squares in all tilings of a k X n rectangle using integer-sided square tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 25, 34, 25, 5, 0, 0, 6, 50, 98, 98, 50, 6, 0, 0, 7, 96, 256, 386, 256, 96, 7, 0, 0, 8, 180, 654, 1402, 1402, 654, 180, 8, 0, 0, 9, 331, 1625, 4938, 6940, 4938, 1625, 331, 9, 0

Offset: 0

Author

Alois P. Heinz, Jun 10 2013

Keywords

• nonn
• tabl

Examples

			A(3,3) = 1 + 6 + 6 + 6 + 6 + 9 = 34:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |     |  |   |_|  |_|   |  |_|_|_|  |_|_|_|  |_|_|_|
  |     |  |___|_|  |_|___|  |_|   |  |   |_|  |_|_|_|
  |_____|  |_|_|_|  |_|_|_|  |_|___|  |___|_|  |_|_|_|
Square array A(n,k) begins:
  0, 0,   0,    0,     0,      0,       0,        0, ...
  0, 1,   2,    3,     4,      5,       6,        7, ...
  0, 2,   5,   12,    25,     50,      96,      180, ...
  0, 3,  12,   34,    98,    256,     654,     1625, ...
  0, 4,  25,   98,   386,   1402,    4938,    16936, ...
  0, 5,  50,  256,  1402,   6940,   33502,   157279, ...
  0, 6,  96,  654,  4938,  33502,  221672,  1426734, ...
  0, 7, 180, 1625, 16936, 157279, 1426734, 12582472, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..30, flattened

Crossrefs

Columns (or rows) k=0-10 give: A000004, A001477, A067331(n-1) for n>0, A226546, A226547, A226548, A226549, A226550, A226551, A226552, A226553.

Main diagonal gives A226554.

Cf. A113881, A219924.

Programs

Maple

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then [0,0] elif n=0 or l=[] then [1,0]
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:=[0$2];
         for i from k to nops(l) while l[i]=0 do s:=s+(h->h+[0, h[1]])
           (b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n]))[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, {0, 0}, n == 0 || l == {}, {1, 0}, Min[l] > 0, t=Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s={0, 0}; For[i=k, i <= Length[l] && l[[i]] == 0, i++, s = s + Function[h, h+{0, h[[1]]}][b[n, Join[l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]]]]] ]; s]]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]][[2]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)