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A226545
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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.
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12
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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)
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OFFSET
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0,8
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LINKS
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns (or rows) k=0-10 give: A000004, A001477, A067331(n-1) for n>0, A226546, A226547, A226548, A226549, A226550, A226551, A226552, A226553.
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KEYWORD
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AUTHOR
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STATUS
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approved
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