A321719 Number of non-normal semi-magic squares with sum of entries equal to n.

1, 1, 3, 7, 28, 121, 746, 5041, 40608, 362936, 3635017, 39916801, 479206146, 6227020801, 87187426839, 1307674521272, 20923334906117, 355687428096001, 6402415241245577, 121645100408832001, 2432905938909013343, 51090942176372298027, 1124001180562929946213

Offset: 0

Author

Gus Wiseman, Nov 18 2018

Keywords

• nonn

Comments

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Squares must be of size k X k where k is a divisor of n. This implies that a(p) = p! + 1 for p prime since the only allowable squares are of sizes 1 X 1 and p X p. The 1 X 1 square is [p], the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 1 for n > 1. - Chai Wah Wu, Jan 13 2019

Examples

			The a(3) = 7 semi-magic squares:
  [3]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100
• Wikipedia, Magic square
• Index entries for sequences related to magic squares

Crossrefs

Cf. A006052, A120732, A120733, A257493, A271103, A319056, A319616.

Cf. A321717, A321718, A321720, A321721, A321722, A321723, A321724.

Programs

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/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

Formula

a(p) = p! + 1 for p prime and a(n) >= n! + 1 for n > 1 (see comment above). - Chai Wah Wu, Jan 13 2019

a(n) = Sum_{d|n} A257493(d, n/d) for n > 0. - Andrew Howroyd, Apr 11 2020

Extensions

a(7) from Chai Wah Wu, Jan 13 2019

a(6) corrected and a(8)-a(15) added by Chai Wah Wu, Jan 14 2019

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

Terms a(20) and beyond from Andrew Howroyd, Apr 11 2020