A097693 Largest achievable determinant of a 3 X 3 matrix whose elements are 9 distinct integers chosen from the range -n...n.

86, 216, 438, 776, 1254, 1896, 2726, 3768, 5046, 6584, 8406, 10536, 12998, 15816, 19014, 22616, 26646, 31128, 36086, 41544, 47526, 54056, 61158, 68856, 77174, 86136, 95766, 106088, 117126, 128904, 141446, 154776, 168918, 183896, 199734

Offset: 4

Author

Hugo Pfoertner, Aug 24 2004

Keywords

• nonn
• easy

Examples

			a(5)=216 because no larger determinant of a 3 X 3 integer matrix b(j,k) with distinct elements -5<=b(j,k)<=5,j=1..3,k=1..3 can be built than det((-5,-4,1),(2,-2,5),(-3,4,3))=216.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Other maximal 3 X 3 determinants: Cf. A097399: 3 X 3 matrix filled with consecutive integers, A097401: 3 X 3 matrix filled with integers from 0...n, A097694, A097695, A097696: corresponding sequences for 4 X 4 matrices.

Formula

An optimal choice and arrangement is of the following form: det((-n, 1-n, n-4), (n-3, 3-n, n), (2-n, n-1, n-2))=2*(2*n^3-7*n^2+6*n+3). There are 35 other equivalent arrangements corresponding to permutations of rows and columns.

G.f.: 2*x^4*(43-64*x+45*x^2-12*x^3)/(1-x)^4. - Colin Barker, Mar 29 2012