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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
Wikipedia, Magic square
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
Sequence in context: A082669 A215574 A146083 * A306208 A001739 A072805
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 March 29 10:22 EDT 2024. Contains 371268 sequences. (Running on oeis4.)