A321723 Number of non-normal magic squares whose entries are all 0 or 1 and sum to n.

1, 1, 0, 0, 9, 20, 96, 656, 5584, 48913, 494264, 5383552, 65103875, 840566080, 11834159652, 176621049784, 2838040416201, 48060623405312

Offset

0,5

Comments

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

Links

Table of n, a(n) for n=0..17.

Wikipedia, Magic square

Index entries for sequences related to magic squares

Formula

a(n) >= A007016(n) with equality if n is prime. - Chai Wah Wu, Jan 15 2019

Example

The a(4) = 9 magic squares:
  [1 1]
  [1 1]
.
  [1 0 0 0][1 0 0 0][0 1 0 0][0 1 0 0][0 0 1 0][0 0 1 0][0 0 0 1][0 0 0 1]
  [0 0 1 0][0 0 0 1][0 0 1 0][0 0 0 1][1 0 0 0][0 1 0 0][1 0 0 0][0 1 0 0]
  [0 0 0 1][0 1 0 0][1 0 0 0][0 0 1 0][0 1 0 0][0 0 0 1][0 0 1 0][1 0 0 0]
  [0 1 0 0][0 0 1 0][0 0 0 1][1 0 0 0][0 0 0 1][1 0 0 0][0 1 0 0][0 0 1 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[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#], SameQ@@Join[{Tr[prs2mat[#]], Tr[Reverse[prs2mat[#]]]}, Total/@prs2mat[#], Total/@Transpose[prs2mat[#]]]]&]], {n, 5}]

Crossrefs

Cf. A006052, A007016, A057151, A068313, A101370, A104602, A120732, A271103, A319056, A319616.

Cf. A321717, A321718, A321719, A321720, A321721, A321722.

Sequence in context: A013573 A146388 A230833 * A282763 A013338 A008847

Adjacent sequences: A321720 A321721 A321722 * A321724 A321725 A321726

Keyword

nonn,more

Author

Gus Wiseman, Nov 18 2018

Extensions

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

a(16)-a(17) from Chai Wah Wu, Jan 16 2019

Status

approved