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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. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

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

EXAMPLE

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, ...

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-Fran├žois Alcover, Dec 13 2013, translated from Maple *)

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.

Sequence in context: A271916 A327031 A014473 * A271917 A185651 A265080

Adjacent sequences:  A226542 A226543 A226544 * A226546 A226547 A226548

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 10 2013

STATUS

approved

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Last modified September 20 22:20 EDT 2019. Contains 327252 sequences. (Running on oeis4.)