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 A220777 Number A(n,k) of tilings of a k X n rectangle using integer sided rectangular tiles of equal area; square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 2, 9, 4, 9, 2, 1, 1, 4, 11, 20, 20, 11, 4, 1, 1, 2, 21, 7, 49, 7, 21, 2, 1, 1, 4, 24, 54, 115, 115, 54, 24, 4, 1, 1, 3, 43, 12, 343, 4, 343, 12, 43, 3, 1, 1, 4, 62, 190, 850, 1225, 1225, 850, 190, 62, 4, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Antidiagonals n = 0..26, flattened EXAMPLE A(3,5) = 7, because there are 7 tilings of a 5 X 3 rectangle using integer sided rectangular tiles of equal area: ._____. ._____. ._____. ._____. ._____. ._____. ._____. |     | | | | | |_____| |_____| |_____| | | | | |_|_|_| |     | | | | | |_____| |_____| | | | | | | | | |_|_|_| |     | | | | | |_____| | | | | | | | | |_|_|_| |_|_|_| |     | | | | | |_____| | | | | |_|_|_| |_____| |_|_|_| |_____| |_|_|_| |_____| |_|_|_| |_____| |_____| |_|_|_| Square array A(n,k) begins: 1,  1,  1,   1,    1,     1,      1,       1,        1, ... 1,  1,  2,   2,    3,     2,      4,       2,        4, ... 1,  2,  4,   6,    9,    11,     21,      24,       43, ... 1,  2,  6,   4,   20,     7,     54,      12,      190, ... 1,  3,  9,  20,   49,   115,    343,     850,     2401, ... 1,  2, 11,   7,  115,     4,   1225,       7,    15242, ... 1,  4, 21,  54,  343,  1225,   7104,   31777,   169952, ... 1,  2, 24,  12,  850,     7,  31777,       4,  1300180, ... 1,  4, 43, 190, 2401, 15242, 169952, 1300180, 13036591, ... MAPLE b:= proc(n, l, d) option remember; local i, k, m, q, s, t;       if max(l[])>n then 0 elif n=0 or l=[] then 1     elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l), d)     else for k do if l[k]=0 then break fi od; s, m:=0, nops(l);          for i from k to m while l[i]=0 do if irem(d, 1+i-k, 'q')=0            and q<=n then s:= s+ b(n, [l[j]\$j=1..k-1, q\$j=k..i,            l[j]\$j=i+1..m], d) fi od; s       fi     end: A:= (n, k)-> `if`(n n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t, d], True, k = Position[l, 0, 1][[1, 1]]; {s, m} = {0, Length[l]}; For[i = k, i <= m && l[[i]] == 0, i++, If[(Mod[d, 1+i-k]) == 0 && (q = Quotient[d, 1+i-k]) <= n, s = s + b[n, Join[l[[1 ;; k-1]], Table[q, {j, k, i}], l[[i+1 ;; m]]], d] ] ]; s ] ]; a[n_, k_] := a[n, k] = If[n < k, a[k, n], If[k == 0, 1, Sum[b[n, Array[0&, k], d], {d, Divisors[n*k]}]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *) CROSSREFS Columns (or rows) k=0-10 give: A000012, A000005, A220768, A220769, A220770, A220771, A220772, A220773, A220774, A220775, A220776. Main diagonal gives: A220778. Sequence in context: A262750 A075402 A276696 * A088855 A034851 A172453 Adjacent sequences:  A220774 A220775 A220776 * A220778 A220779 A220780 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 19 2012 STATUS approved

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Last modified July 30 09:18 EDT 2021. Contains 346359 sequences. (Running on oeis4.)