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A321722 Number of non-normal magic squares whose entries are nonnegative integers summing to n. 13
1, 1, 1, 1, 10, 21, 97, 657, 5618, 48918, 494530, 5383553, 65112565, 840566081, 11834555867, 176621056393, 2838064404989, 48060623405313 (list; graph; refs; listen; history; text; internal format)
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(p) = A007016(p) + 1 if p is prime. a(n) >= A007016(n) + 1 for n > 1. - Chai Wah Wu, Jan 15 2019

EXAMPLE

The a(4) = 10 magic squares:

  [4]

.

  [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[multsubs[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, A120732, A271103, A319056, A319616.

Cf. A321718, A321719, A321720, A321721.

Sequence in context: A082669 A215574 A146083 * A306208 A001739 A072805

Adjacent sequences:  A321719 A321720 A321721 * A321723 A321724 A321725

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

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Last modified August 4 10:11 EDT 2021. Contains 346447 sequences. (Running on oeis4.)