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.

A374239 a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 1, 0, -1, 1, -545, -13805, -301184, -18551951, -352513176, -31451535983, -1209153784888, -87868166035113, -4204963833160760, -664087819207293468
Offset: 0

Views

Author

Stefano Spezia, Jul 01 2024

Keywords

Examples

			a(5) = -545:
  [1, 1, 4, 2, 3]
  [1, 1, 1, 4, 2]
  [4, 1, 1, 1, 4]
  [2, 4, 1, 1, 1]
  [3, 2, 4, 1, 1]
		

Crossrefs

Cf. A374240 (maximal), A374241 (maximal absolute value), A374242 (minimal nonzero absolute value).

Programs

  • Mathematica
    a[0]=1; a[n_]:=Min[Table[Det[ToeplitzMatrix[Join[{1}, Part[Permutations[Range[n - 1]], i]]]], {i, (n-1)!}]]; Array[a, 11, 0]

Extensions

a(11)-a(14) from Lucas A. Brown, Oct 10 2024

A374240 a(n) is the maximal determinant of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 1, 0, 0, 40, 256, 12232, 526773, 8684025, 768561384, 28938090375, 1273675677456, 73821863714933, 7601760995500947, 527066887623562528
Offset: 0

Views

Author

Stefano Spezia, Jul 01 2024

Keywords

Examples

			a(5) = 256:
  [1, 4, 1, 2, 3]
  [4, 1, 4, 1, 2]
  [1, 4, 1, 4, 1]
  [2, 1, 4, 1, 4]
  [3, 2, 1, 4, 1]
		

Crossrefs

Cf. A374239 (minimal), A374241 (maximal absolute value), A374242 (minimal nonzero absolute value).

Programs

  • Mathematica
    a[0]=1; a[n_]:=Max[Table[Det[ToeplitzMatrix[Join[{1}, Part[Permutations[Range[n - 1]], i]]]], {i, (n-1)!}]]; Array[a, 11, 0]

Extensions

a(11)-a(14) from Lucas A. Brown, Oct 10 2024

A374242 a(n) is the minimal absolute value of the determinant of a nonsingular n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 1, 3, 9, 3, 1, 5, 9, 1, 1, 1, 1
Offset: 3

Views

Author

Stefano Spezia, Jul 01 2024

Keywords

Comments

The offset is 3 because for n = 2 the unique symmetric Toeplitz matrix having 1 on the main diagonal and 1 off-diagonal is singular.
Conjecture: all the terms are odd.

Examples

			a(5) = 3:
  [1, 1, 2, 3, 4]
  [1, 1, 1, 2, 3]
  [2, 1, 1, 1, 2]
  [3, 2, 1, 1, 1]
  [4, 3, 2, 1, 1]
		

Crossrefs

Cf. A374239 (minimal), A374240 (maximal), A374241 (maximal absolute value).

Programs

  • Mathematica
    a[n_]:=Min[Select[Table[Abs[Det[ToeplitzMatrix[Join[{1}, Part[Permutations[Range[n - 1]], i]]]]], {i, (n-1)!}],Positive]]; Array[a, 8, 3]

Extensions

a(11)-a(14) from Lucas A. Brown, Oct 10 2024

A374342 a(n) is the maximal absolute value of the determinant of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the first n-1 primes off-diagonal.

Original entry on oeis.org

1, 1, 3, 15, 259, 12167, 1708047, 157042600, 33081320935, 3336975844504, 579469550006151
Offset: 0

Views

Author

Stefano Spezia, Jul 05 2024

Keywords

Examples

			a(5) = 12167:
  [1, 2, 7, 3, 5]
  [2, 1, 2, 7, 3]
  [7, 2, 1, 2, 7]
  [3, 7, 2, 1, 2]
  [5, 3, 7, 2, 1]
		

Crossrefs

Cf. A374340 (minimal), A374341 (maximal), A374343 (minimal nonzero absolute value), A374067 (minimal permanent), A374345 (maximal permanent).

Programs

  • Mathematica
    a[n_]:=Max[Table[Abs[Det[ToeplitzMatrix[Join[{1},Part[Permutations[Prime[Range[n-1]]],i]]]]],{i,(n-1)!}]]; Join[{1},Array[a,10]]

Formula

a(n) = max(abs(A374340(n)),A374341(n)).

A374617 a(n) is the number of distinct values of the determinant of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 1, 1, 2, 6, 21, 111, 710, 4968, 39879, 360952
Offset: 0

Views

Author

Stefano Spezia, Jul 14 2024

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_]:=CountDistinct[Table[Det[ToeplitzMatrix[Join[{1},Part[Permutations[Range[n-1]], i]]]], {i, (n -1)!}]]; Join[{1}, Array[a, 10]]

Formula

a(n) <= (n-1)! for n > 0.

A374278 a(n) is the maximal permanent of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 1, 2, 18, 389, 14284, 798322, 62490160, 6519511313, 873036867840, 145856387327074
Offset: 0

Views

Author

Stefano Spezia, Jul 02 2024

Keywords

Examples

			a(5) = 14284:
  [1, 4, 3, 2, 1]
  [4, 1, 4, 3, 2]
  [3, 4, 1, 4, 3]
  [2, 3, 4, 1, 4]
  [1, 2, 3, 4, 1]
		

Crossrefs

Programs

  • Mathematica
    a[0]=1; a[n_]:=Max[Table[Permanent[ToeplitzMatrix[Join[{1}, Part[Permutations[Range[n - 1]], i]]]], {i, (n-1)!}]]; Array[a, 11, 0]
Showing 1-6 of 6 results.