cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

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Author

Hugo Pfoertner, Aug 24 2004

Keywords

Examples

			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.

Links

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.

Programs

  • 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
  • Mathematica
    LinearRecurrence[{4,-6,4,-1},{332,528,796,1148},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

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

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

Views

Author

Hugo Pfoertner, Aug 24 2004

Keywords

Examples

			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.

Links

Crossrefs

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.

Formula

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

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

Views

Author

Hugo Pfoertner, Aug 24 2004

Keywords

Examples

			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.

Crossrefs

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.

Formula

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

Views

Author

Hugo Pfoertner, Aug 25 2004

Keywords

Comments

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

Examples

			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.

Crossrefs

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.

Formula

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

Views

Author

Hugo Pfoertner, Aug 25 2004

Keywords

Crossrefs

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).

Formula

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]

A097400 Number of different values that can be assumed by the determinant of a 3 X 3 matrix whose elements are all permutations of the consecutive integers in the range (n-4,n+4).

Original entry on oeis.org

167, 207, 341, 489, 635, 777, 913, 1055, 1163, 1325, 1389, 1573, 1643, 1819, 1867, 2039, 2073, 2229, 2295, 2463, 2471, 2649, 2625, 2843, 2787, 2995, 2917, 3171, 3099, 3309, 3241, 3397, 3379, 3557, 3467, 3707, 3631, 3807, 3731, 3919, 3869, 3999, 3947
Offset: 0

Views

Author

Hugo Pfoertner, Aug 19 2004

Keywords

Comments

a(j)=5761 for all j>=167.

Examples

			a(0)=167 because the only determinants not achievable by permuting the positions of the matrix elements -4...+4 are +-81,+-83 and +-85, i.e. a(0)=2*A097399(0)-6+1=167 (+1 for det=0).
a(1)=207 because all values -102...+102 and +-104 can be represented as determinant of a matrix whose elements are a permutation of the 9 numbers -3...5.

Crossrefs

Cf. A097399 corresponding maximum determinant, a(5)=A088217(3).
Showing 1-6 of 6 results.

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