cp's OEIS Frontend

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

%I A097401 #23 Sep 08 2022 08:45:14
%S A097401 332,528,796,1148,1596,2152,2828,3636,4588,5696,6972,8428,10076,11928,
%T A097401 13996,16292,18828,21616,24668,27996,31612,35528,39756,44308,49196,
%U A097401 54432,60028,65996,72348,79096,86252,93828,101836,110288,119196,128572
%N A097401 Largest achievable determinant of a 3 X 3 matrix whose elements are 9 distinct nonnegative integers chosen from the range 0..n.
%H A097401 Vincenzo Librandi, <a href="/A097401/b097401.txt">Table of n, a(n) for n = 8..1000</a>
%H A097401 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A097401 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.
%F A097401 a(n) = 2*n^3 - 18*n^2 + 68*n - 84.
%F A097401 G.f.: 4*x^8*(83 - 200*x + 169*x^2 - 49*x^3)/(1-x)^4. - _Colin Barker_, Mar 29 2012
%F A097401 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 25 2012
%e A097401 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.
%t A097401 LinearRecurrence[{4,-6,4,-1},{332,528,796,1148},40] (* _Vincenzo Librandi_, Jun 25 2012 *)
%o A097401 (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
%o A097401 (PARI) a(n)=2*n^3-18*n^2+68*n-84 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A097401 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.
%K A097401 nonn,easy
%O A097401 8,1
%A A097401 _Hugo Pfoertner_, Aug 24 2004