A097694 -id:A097694 - 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.

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

Hugo Pfoertner, Aug 24 2004

			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.

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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.

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

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

a(n)=2*n^3-18*n^2+68*n-84 \\ Charles R Greathouse IV, Oct 07 2015

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

A097693 Largest achievable determinant of a 3 X 3 matrix whose elements are 9 distinct integers chosen from the range -n...n.

Original entry on oeis.org

86, 216, 438, 776, 1254, 1896, 2726, 3768, 5046, 6584, 8406, 10536, 12998, 15816, 19014, 22616, 26646, 31128, 36086, 41544, 47526, 54056, 61158, 68856, 77174, 86136, 95766, 106088, 117126, 128904, 141446, 154776, 168918, 183896, 199734

Offset: 4

Hugo Pfoertner, Aug 24 2004

			a(5)=216 because no larger determinant of a 3 X 3 integer matrix b(j,k) with distinct elements -5<=b(j,k)<=5,j=1..3,k=1..3 can be built than det((-5,-4,1),(2,-2,5),(-3,4,3))=216.

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

Other maximal 3 X 3 determinants: Cf. A097399: 3 X 3 matrix filled with consecutive integers, A097401: 3 X 3 matrix filled with integers from 0...n, A097694, A097695, A097696: corresponding sequences for 4 X 4 matrices.

An optimal choice and arrangement is of the following form: det((-n, 1-n, n-4), (n-3, 3-n, n), (2-n, n-1, n-2))=2*(2*n^3-7*n^2+6*n+3). There are 35 other equivalent arrangements corresponding to permutations of rows and columns.

G.f.: 2*x^4*(43-64*x+45*x^2-12*x^3)/(1-x)^4. - Colin Barker, Mar 29 2012

A097695 Largest achievable determinant of a 4 X 4 matrix whose elements are 16 distinct integers chosen from the range -n...n.

Original entry on oeis.org

10324, 19920, 35791, 60122, 95610, 145362

Offset: 8

Hugo Pfoertner, Aug 25 2004

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

			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.

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.

A097696 Largest achievable determinant of a 4 X 4 matrix whose elements are the 16 consecutive integers n-15,...,n.

Original entry on oeis.org

7343, 8784, 12065, 16800, 21600, 26400, 31200, 36000, 40800, 45600, 50400, 55200, 60000, 64800, 69600, 74400, 79200, 84000, 88800, 93600, 98400, 103200, 108000, 112800, 117600, 122400, 127200, 132000, 136800, 141600, 146400, 151200, 156000

Offset: 8

Hugo Pfoertner, Aug 25 2004

nonn

Other maximal 4 X 4 determinants: Cf. A097694: 4 X 4 matrix filled with integers from 0...n, A097695: 4 X 4 matrix filled with integers from -n...n. A097399, A097401, A097693: corresponding sequences for 3 X 3 matrices. a(16)=A085000(4).

For n>10 an arrangement maximizing the determinant is of the following form: det((n, n-9, n-13, n-8), (n-12, n-1, n-11, n-5), (n-7, n-6, n-2, n-15), (n-10, n-14, n-4, n-3)) =2400*(2*n-15). a(n)=a(15-n) for n<8.

Empirical G.f.: x^8*(65*x^4+1454*x^3+1840*x^2-5902*x+7343) / (x-1)^2. [Colin Barker, Jan 10 2013]

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

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A097693 Largest achievable determinant of a 3 X 3 matrix whose elements are 9 distinct integers chosen from the range -n...n.

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A097695 Largest achievable determinant of a 4 X 4 matrix whose elements are 16 distinct integers chosen from the range -n...n.

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A097696 Largest achievable determinant of a 4 X 4 matrix whose elements are the 16 consecutive integers n-15,...,n.

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