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A220122
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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.
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16
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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
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OFFSET
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0,13
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COMMENTS
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Row n gives: 1 followed by period A003418(n): (1, A000045(n+1), ...) repeated; offset 0.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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