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A321718 Number of coupled non-normal semi-magic rectangles with sum of entries equal to n. 13
1, 1, 5, 9, 44, 123, 986, 5043, 45832, 366300, 3862429, 39916803, 495023832, 6227020803, 88549595295, 1308012377572, 21086922542349, 355687428096003, 6427700493998229, 121645100408832003 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..19.

Wikipedia, Magic square

FORMULA

a(p) = p! + 3 for p prime. a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019

EXAMPLE

The a(3) = 9 coupled semi-magic rectangles:

  [3] [1 1 1]

.

  [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]

MATHEMATICA

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}]

CROSSREFS

Cf. A006052, A120733, A271103, A319056, A321654.

Cf. A321717, A321719, A321720, A321721, A321722, A321724, A321724.

Sequence in context: A176751 A123822 A226065 * A304127 A220518 A145031

Adjacent sequences:  A321715 A321716 A321717 * A321719 A321720 A321721

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 18 2018

EXTENSIONS

a(7)-a(15) from Chai Wah Wu, Jan 15 2019

a(16)-a(19) from Chai Wah Wu, Jan 16 2019

STATUS

approved

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Last modified June 16 14:16 EDT 2021. Contains 345057 sequences. (Running on oeis4.)