A097401 Largest achievable determinant of a 3 X 3 matrix whose elements are 9 distinct nonnegative integers chosen from the range 0..n.

332, 528, 796, 1148, 1596, 2152, 2828, 3636, 4588, 5696, 6972, 8428, 10076, 11928, 13996, 16292, 18828, 21616, 24668, 27996, 31612, 35528, 39756, 44308, 49196, 54432, 60028, 65996, 72348, 79096, 86252, 93828, 101836, 110288, 119196, 128572

Offset

8,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 8..1000

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

Formula

An optimal choice and arrangement is of the following form: det((n, n-5, 1), (2, n-1, n-3), (n-4, 0, n-2)) = 2*(n^3 - 9*n^2 + 34*n - 42). There are 35 other equivalent arrangements corresponding to permutations of rows and columns.

a(n) = 2*n^3 - 18*n^2 + 68*n - 84.

G.f.: 4*x^8*(83 - 200*x + 169*x^2 - 49*x^3)/(1-x)^4. - Colin Barker, Mar 29 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 25 2012

Example

a(10)=796 because no larger determinant of a 3 X 3 matrix b(j,k) with distinct elements 0 <= b(j,k) <= 10, j=1..3, k=1..3 can be built than det((10,5,1), (2,9,7), (6,0,8)) = 796.

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {332, 528, 796, 1148}, 40] (* Vincenzo Librandi, Jun 25 2012 *)

Prog

(Magma) I:=[332, 528, 796, 1148]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jun 25 2012
(PARI) a(n)=2*n^3-18*n^2+68*n-84 \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Other maximal 3 X 3 determinants: Cf. a(8)=A097399(4)=332: 3 X 3 matrix filled with consecutive integers, A097693: 3 X 3 matrix filled with integers from -n...n, A097694, A097695, A097696: corresponding sequences for 4 X 4 matrices.

Sequence in context: A255389 A133141 A231755 * A250758 A114084 A257892

Adjacent sequences: A097398 A097399 A097400 * A097402 A097403 A097404

Keyword

nonn,easy

Author

Hugo Pfoertner, Aug 24 2004

Status

approved