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 A220122 Number A(n,k) of tilings of a k X n rectangle using integer sided rectangular tiles of area k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 16
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 5, 1, 1, 1, 1, 1, 3, 3, 8, 1, 1, 1, 1, 2, 1, 9, 4, 13, 1, 1, 1, 1, 1, 4, 1, 16, 6, 21, 1, 1, 1, 1, 2, 1, 7, 2, 35, 9, 34, 1, 1, 1, 1, 1, 3, 1, 13, 3, 65, 13, 55, 1, 1, 1, 1, 2, 2, 9, 1, 46, 4, 143, 19, 89, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS Row n gives: 1 followed by period A003418(n): (1, A000045(n+1), ... ) repeated; offset 0. LINKS Alois P. Heinz, Antidiagonals n = 0..32, flattened FORMULA For prime p column p has g.f.: 1/(1-x-x^p) or a_p(n) = Sum_{j=0..floor(n/p)} C(n-(p-1)*j,j). EXAMPLE A(4,4) = 9, because there are 9 tilings of a 4 X 4 rectangle using integer sided rectangular tiles of area 4: ._._._._.  ._______.  .___.___.  ._.___._.  ._______. | | | | |  |_______|  |   |   |  | |   | |  |_______| | | | | |  |_______|  |___|___|  | |___| |  |   |   | | | | | |  |_______|  |   |   |  | |   | |  |___|___| |_|_|_|_|  |_______|  |___|___|  |_|___|_|  |_______| ._._.___.  ._______.  .___._._.  .___.___. | | |   |  |_______|  |   | | |  |   |   | | | |___|  |_______|  |___| | |  |___|___| | | |   |  |   |   |  |   | | |  |_______| |_|_|___|  |___|___|  |___|_|_|  |_______| Square array A(n,k) begins: 1, 1,  1,  1,   1, 1,    1, 1,    1,  1,   1, ... 1, 1,  1,  1,   1, 1,    1, 1,    1,  1,   1, ... 1, 1,  2,  1,   2, 1,    2, 1,    2,  1,   2, ... 1, 1,  3,  2,   3, 1,    4, 1,    3,  2,   3, ... 1, 1,  5,  3,   9, 1,    7, 1,    9,  3,   5, ... 1, 1,  8,  4,  16, 2,   13, 1,   16,  4,   9, ... 1, 1, 13,  6,  35, 3,   46, 1,   35,  6,  15, ... 1, 1, 21,  9,  65, 4,   88, 2,   65,  9,  26, ... 1, 1, 34, 13, 143, 5,  209, 3,  250, 13,  44, ... 1, 1, 55, 19, 281, 6,  473, 4,  495, 37,  75, ... 1, 1, 89, 28, 590, 8, 1002, 5, 1209, 64, 254, ... MAPLE b:= proc(n, l) option remember; local i, k, m, 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))     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(m, 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]) fi od; s       fi     end: A:= (n, k)-> b(n, [0\$k]): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA b[n_, l_] := b[n, l] = Module[{i, k, m, s, t}, Which[Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; {s, m} = {0, Length[l]}; For[ i = k , i <= m && l[[i]] == 0, i++, If[Mod[m, 1+i-k ] == 0 && (q = Quotient[m, 1+i-k]) <= n, s = s+b[n, Join[ l[[1 ;; k-1]], Array[q &, i-k+1], l[[i+1 ;; m]] ]]]]; s]]; a[n_, k_] := b[n, Array[0&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *) CROSSREFS Columns k=0+1, 2-11, 13 give: A000012, A000045(n+1), A000930, A220123, A003520, A220124, A005709, A220125, A220126, A220127, A017905(n+11), A017907(n+13). Rows n=0+1, 2-10 give: A000012, A040001, A220128, A220129, A220130, A220131, A220132, A220133, A220134, A220135. Diagonal gives: A182106. Sequence in context: A033630 A308608 A323719 * A101446 A333769 A259396 Adjacent sequences:  A220119 A220120 A220121 * A220123 A220124 A220125 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 05 2012 STATUS approved

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Last modified April 17 02:26 EDT 2021. Contains 343059 sequences. (Running on oeis4.)