A097693 -id:A097693 - cp's OEIS frontend

A097399 Maximum of the determinant over all permutations of the entries of a 3 X 3 matrix which are consecutive integers in the range (n-4,n+4).

86, 104, 172, 252, 332, 412, 492, 572, 652, 732, 812, 892, 972, 1053, 1134, 1215, 1296, 1377, 1458, 1539, 1620, 1701, 1782, 1863, 1944, 2025, 2106, 2187, 2268, 2349, 2430, 2511, 2592, 2673, 2754, 2835, 2916, 2997, 3078, 3159, 3240, 3321, 3402, 3483, 3564

Offset: 0

Hugo Pfoertner, Aug 19 2004

nonn

			a(0)=86 because the maximal determinant that can achieved using the consecutive integers -4,-3,-2,-1,0,1,2,3,4 as matrix elements of a 3 X 3 matrix is det((-4,-3,0),(1,-1,4),(-2,3,2))=86. Another example for a(5)=412 is given in A085000.

Cf. A097400 = corresponding number of different determinants, A097401, A097693 = maximum of determinant if distinct matrix elements are selected from given range, a(5)=A085000(3) maximal determinant with elements (1..n^2).

Join[{86,104,172,252,332,412,492,572,652,732,812,892},LinearRecurrence[ {2,-1},{972,1053},40]] (* or *) Table[ Det[ Partition[ #,3]]&/@ Permutations[ Range[n-4,n+4]]//Max,{n,0,45}] (* Harvey P. Dale, Jan 14 2015 *)

G.f.: (x^13+12*x^3+50*x^2-68*x+86) / (x-1)^2. [Colin Barker, Dec 13 2012] [I suspect this is merely a conjecture. - N. J. A. Sloane, Jun 09 2018]

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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

Offset: 15

Hugo Pfoertner, Aug 24 2004

			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.

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.

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.

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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A097399 Maximum of the determinant over all permutations of the entries of a 3 X 3 matrix which are consecutive integers in the range (n-4,n+4).

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