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A097693 Largest achievable determinant of a 3 X 3 matrix whose elements are 9 distinct integers chosen from the range -n...n. 5

%I #6 Mar 31 2012 10:29:05

%S 86,216,438,776,1254,1896,2726,3768,5046,6584,8406,10536,12998,15816,

%T 19014,22616,26646,31128,36086,41544,47526,54056,61158,68856,77174,

%U 86136,95766,106088,117126,128904,141446,154776,168918,183896,199734

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

%F 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.

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

%e Example: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

%e det((-5,-4,1),(2,-2,5),(-3,4,3))=216.

%Y 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.

%K nonn,easy

%O 4,1

%A _Hugo Pfoertner_, Aug 24 2004

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