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A321718
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Number of coupled non-normal semi-magic rectangles with sum of entries equal to n.
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13
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1, 1, 5, 9, 44, 123, 986, 5043, 45832, 366300, 3862429, 39916803, 495023832, 6227020803, 88549595295, 1308012377572, 21086922542349, 355687428096003, 6427700493998229, 121645100408832003
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OFFSET
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0,3
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COMMENTS
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A coupled non-normal semi-magic rectangle is a nonnegative integer matrix with equal row sums and equal column sums. The common row sum may be different from the common column sum.
Rectangles must be of size k X m where k and m are divisors of n. This implies that a(p) = p! + 3 for p prime since the only allowable rectangles are of sizes 1 X 1, 1 X p, p X 1 and p X p. The 1 X 1 square is [p], the 1 X p and p X 1 rectangles are [1,...,1] and its transpose and the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019
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LINKS
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FORMULA
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a(p) = p! + 3 for p prime. a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019
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EXAMPLE
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The a(3) = 9 coupled semi-magic rectangles:
[3] [1 1 1]
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[1] [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
[1] [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
[1] [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
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MATHEMATICA
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prs2mat[prs_]:=Table[Count[prs, {i, j}], {i, Union[First/@prs]}, {j, Union[Last/@prs]}];
multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n], 2], n], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], SameQ@@Total/@prs2mat[#], SameQ@@Total/@Transpose[prs2mat[#]]]&]], {n, 5}]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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