

A097401


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


6



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
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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, n5, 1), (2, n1, n3), (n4, 0, n2)) = 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)/(1x)^4.  Colin Barker, Mar 29 2012
a(n) = 4*a(n1)  6*a(n2) + 4*a(n3)  a(n4).  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(n1)6*Self(n2)+4*Self(n3)Self(n4): n in [1..40]]; // Vincenzo Librandi, Jun 25 2012
(PARI) a(n)=2*n^318*n^2+68*n84 \\ 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



