8,1

The formula for a(12) and a(13) gives lower bounds for the next terms a(14)>=212802, a(15)>=301770.

Optimal choices and arrangements: n=8 see example, n=9, 10, 11: det((n, n-1, 7-n, n-5), (5-n, n-4, 1-n, 4-n), (n-2, 2-n, 3-n, 6-n), (n-6, n-7, n-3, -n))= 16*n^4-224*n^3+1334*n^2-3795*n+4341. n=12, 13: det((n, n-5, n-3, 6-n), (n-6, -n, 5-n, 3-n), (7-n, n-1, 4-n, 2-n), (n-2, n-7, 1-n, n-4))= 2*(8*n^4-112*n^3+670*n^2-1947*n+2325). For each n there are (4!)^2=576 equivalent arrangements corresponding to permutations of rows and columns.

a(8)=10324 because no larger determinant of a 4 X 4 integer matrix b(j,k) with distinct elements -8<=b(j,k)<=8,j=1..4,k=1..4 can be built than

det((8,4,3,2),(-1,7,-4,-6),(5,-5,-7,-2),(1,-3,6,-8))=10324.

Other maximal 4 X 4 determinants: Cf. A097694: 4 X 4 matrix filled with integers from 0...n, A097696: 4 X 4 matrix filled with consecutive integers. A097399, A097401, A097693: corresponding sequences for 3 X 3 matrices.

Sequence in context: A099746 A083438 A144125 this_sequence A031803 A035912 A104439

Adjacent sequences: A097692 A097693 A097694 this_sequence A097696 A097697 A097698

more,nonn

Hugo Pfoertner (hugo(AT)pfoertner.org), Aug 25 2004