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A245013
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Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and 2 X 2 squares; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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12
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 5, 5, 1, 1, 1, 1, 8, 11, 11, 8, 1, 1, 1, 1, 13, 21, 35, 21, 13, 1, 1, 1, 1, 21, 43, 93, 93, 43, 21, 1, 1, 1, 1, 34, 85, 269, 314, 269, 85, 34, 1, 1, 1, 1, 55, 171, 747, 1213, 1213, 747, 171, 55, 1, 1
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OFFSET
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0,13
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LINKS
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EXAMPLE
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A(3,3) = 5:
._._._. .___._. ._.___. ._._._. ._._._.
|_|_|_| | |_| |_| | |_|_|_| |_|_|_|
|_|_|_| |___|_| |_|___| |_| | | |_|
|_|_|_| |_|_|_| |_|_|_| |_|___| |___|_| .
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 3, 5, 8, 13, 21, ...
1, 1, 3, 5, 11, 21, 43, 85, ...
1, 1, 5, 11, 35, 93, 269, 747, ...
1, 1, 8, 21, 93, 314, 1213, 4375, ...
1, 1, 13, 43, 269, 1213, 6427, 31387, ...
1, 1, 21, 85, 747, 4375, 31387, 202841, ...
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MAPLE
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b:= proc(n, l) option remember; local m, k; m:= min(l[]);
if m>0 then b(n-m, map(x->x-m, l))
elif n=0 then 1
else for k while l[k]>0 do od; b(n, subsop(k=1, l))+
`if`(n>1 and k<nops(l) and l[k+1]=0,
b(n, subsop(k=2, k+1=2, l)), 0)
fi
end:
A:= (n, k)-> `if`(min(n, k)<2, 1, b(max(n, k), [0$min(n, 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[{m=Min[l], k}, If[m>0, b[n-m, l-m], If[n == 0, 1, k=Position[l, 0, 1, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[n>1 && k<Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0]]]]; A[n_, k_] := If[Min[n, k]<2, 1, b[Max[n, k], Table[0, {Min[n, k]}]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2014, after Alois P. Heinz *)
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CROSSREFS
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Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A001045(n+1), A054854, A054855, A063650, A063651, A063652, A063653, A063654.
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KEYWORD
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AUTHOR
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STATUS
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approved
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