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MATHEMATICA
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(* The code is in the InputForm form to simply copy and paste it in Mathematica. The input parameter is n>=1 (odd) and for larger n's the code must be preceded by: *)
SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->1000}];
(* For a very large n the parameter value (1000) must be increased further but the enumeration is increasingly time-consuming. *)
Reduce[Subscript[a, 1, 2]+Subscript[a, 1, 3]+Subscript[a, 1, 4]==n-2*Subscript[a, 1, 1]&&Subscript[a, 1, 2]>=0&&Subscript[a, 1, 3]>=0&&Subscript[a, 1, 4]>=0&&Subscript[a, 1, 1]>=0&&Subscript[a, 1, 2]+Subscript[a, 2, 3]+Subscript[a, 2, 4]==n-2*Subscript[a, 2, 2]&&Subscript[a, 2, 3]>=0&&Subscript[a, 2, 4]>=0&&Subscript[a, 2, 2]>=0&&Subscript[a, 1, 3]+Subscript[a, 2, 3]+Subscript[a, 3, 4]==n-2*Subscript[a, 3, 3]&&Subscript[a, 3, 4]>=0&&Subscript[a, 3, 3]>=0&&Subscript[a, 1, 4]+Subscript[a, 2, 4]+Subscript[a, 3, 4]==n-2*Subscript[a, 4, 4]&&Subscript[a, 4, 4]>=0, {Subscript[a, 1, 1], Subscript[a, 1, 2], Subscript[a, 1, 3], Subscript[a, 1, 4], Subscript[a, 2, 2], Subscript[a, 2, 3], Subscript[a, 2, 4], Subscript[a, 3, 3], Subscript[a, 3, 4], Subscript[a, 4, 4]}, Integers]//Length
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