A097694 Largest achievable determinant of a 4 X 4 matrix whose elements are 16 distinct nonnegative integers chosen from the range 0...n.

36000, 50736, 69828, 94092, 124699, 162604, 208697, 264094, 329983, 407624, 498349, 603562

Offset

15,1

Links

Table of n, a(n) for n=15..26.

Formula

For n<=18 an optimal choice and arrangement is of the following form det((n, 6, 2, n-8), (3, n-1, 4, n-5), (n-7, n-6, n-2, 0), (5, 1, n-4, n-3))= 3*n^4-62*n^3+543*n^2-2128*n+3120. For n>18 up to the limit investigated (n<=26) one choice maximizing the determinant is det((n, 6, 2, n-8), (3, n-1, 4, n-5), (n-7, n-6, n-3, 0), (5, 1, n-2, n-4))= 3*n^4-60*n^3+491*n^2-1821*n+2624. In both cases there are 575 other equivalent arrangements corresponding to permutations of rows and columns.

Example

a(18)=94092 because no 4 X 4 matrix b(j,k) with distinct elements 0<=b(j,k)<=18,j=1..4,k=1..4 can be built that has a larger determinant than
det((18,6,2,10),(3,17,4,13),(11,12,16,0),(5,1,14,15))=94092.

Crossrefs

Other maximal 4 X 4 determinants: Cf. A097696: 4 X 4 matrix filled with consecutive integers, A097695: 4 X 4 matrix filled with integers from -n...n, A097399, A097401, A097693: corresponding sequences for 3 X 3 matrices, A085000: n X n matrix filled with consecutive integers 1...n^2.

Sequence in context: A236201 A034605 A031827 * A209956 A256365 A011766

Adjacent sequences: A097691 A097692 A097693 * A097695 A097696 A097697

Keyword

more,nonn

Author

Hugo Pfoertner, Aug 24 2004

Status

approved