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 1-7 of 7 results.

A350954 Maximal determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.

Original entry on oeis.org

1, 1, 3, 15, 100, 3091, 49375, 1479104, 43413488, 1539619328, 64563673460, 2877312739624, 252631974548628
Offset: 0

Views

Author

Stefano Spezia, Jan 27 2022

Keywords

Examples

			a(3) = 15:
    1    3    2
    3    1    3
    2    3    1
a(4) = 100:
    2    1    4    3
    1    2    1    4
    4    1    2    1
    3    4    1    2
a(5) = 3091:
    3    5    1    2    4
    5    3    5    1    2
    1    5    3    5    1
    2    1    5    3    5
    4    2    1    5    3
		

Crossrefs

Cf. A307887, A350931, A350953 (minimal), A356865 (minimal nonzero absolute value).

Programs

  • Python
    from itertools import permutations
    from sympy import Matrix
    def A350954(n): return max(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).det() for p in permutations(range(1,n+1))) # Chai Wah Wu, Jan 27 2022

Extensions

a(9) from Alois P. Heinz, Jan 27 2022
a(10)-a(12) from Lucas A. Brown, Sep 01 2022

A350953 Minimal determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.

Original entry on oeis.org

1, 1, -3, -12, -100, -1749, -47600, -800681, -39453535, -1351201968, -66984136299, -2938096403400, -235011452211680
Offset: 0

Views

Author

Stefano Spezia, Jan 27 2022

Keywords

Examples

			a(3) = -12:
    2    3    1
    3    2    3
    1    3    2
a(4) = -100:
    3    4    1    2
    4    3    4    1
    1    4    3    4
    2    1    4    3
a(5) = -1749:
    5    4    1    3    2
    4    5    4    1    3
    1    4    5    4    1
    3    1    4    5    4
    2    3    1    4    5
		

Crossrefs

Cf. A307887, A350930, A350954 (maximal), A356865 (minimal nonzero absolute value).

Programs

  • Python
    from itertools import permutations
    from sympy import Matrix
    def A350953(n): return min(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).det() for p in permutations(range(1,n+1))) # Chai Wah Wu, Jan 27 2022

Extensions

a(9) from Alois P. Heinz, Jan 27 2022
a(10)-a(12) from Lucas A. Brown, Sep 01 2022

A369830 a(n) is the number of distinct values of the determinant of an n X n symmetric Toeplitz matrix using the integers 1 to n.

Original entry on oeis.org

1, 1, 2, 5, 22, 97, 613, 4749, 38493, 353684
Offset: 0

Views

Author

Stefano Spezia, Feb 03 2024

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := CountDistinct[Table[Det[ToeplitzMatrix[Part[Permutations[Range[n]], i]]], {i, n !}]]; Join[{1}, Array[a,9]]
  • Python
    from itertools import permutations
    from sympy import Matrix
    def A369830(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).det() for p in permutations(range(1,n+1))}) # Chai Wah Wu, Feb 12 2024

Formula

a(n) <= A000142(n).

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

A358325 a(n) is the minimal absolute value of determinant of a nonsingular n X n symmetric Toeplitz matrix using the integers 0 to n - 1.

Original entry on oeis.org

1, 3, 12, 2, 11, 10, 5, 4, 1, 4, 1
Offset: 2

Views

Author

Stefano Spezia, Nov 09 2022

Keywords

Examples

			a(3) = 3:
    [1, 0, 2;
     0, 1, 0;
     2, 0, 1]
a(4) = 12:
    [3, 2, 1, 0;
     2, 3, 2, 1;
     1, 2, 3, 2;
     0, 1, 2, 3]
a(5) = 2:
    [0, 4, 1, 2, 3;
     4, 0, 4, 1, 2;
     1, 4, 0, 4, 1;
     2, 1, 4, 0, 4;
     3, 2, 1, 4, 0]
		

Crossrefs

Cf. A356865.
Cf. A358323 (minimal signed), A358324 (maximal signed), A358326 (minimal permanent), A358327 (maximal permanent).

Programs

  • Mathematica
    Table[Min[Select[Table[Abs[Det[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n-1]]], i]]]], {i, n!}], Positive]], {n, 2, 9}]  (* Corrected by Stefano Spezia, Nov 17 2022 *)

Extensions

a(3), a(7), and a(10)-a(12) from Lucas A. Brown, Nov 16 2022

A364230 Triangle read by rows: T(n, k) is the number of n X n symmetric Toeplitz matrices of rank k using all the integers 1, 2, ..., n.

Original entry on oeis.org

1, 0, 2, 0, 0, 6, 0, 0, 0, 24, 0, 0, 0, 0, 120, 0, 0, 0, 0, 2, 718, 0, 0, 0, 0, 4, 31, 5005, 0, 0, 0, 0, 0, 2, 44, 40274, 0, 0, 0, 0, 0, 0, 4, 272, 362604, 0, 0, 0, 0, 0, 0, 0, 111, 774, 3627915, 0, 0, 0, 0, 0, 0, 2, 14, 244, 6974, 39909566, 0, 0, 0, 0, 0, 0, 0, 4, 64, 743, 9533, 478991256
Offset: 1

Views

Author

Stefano Spezia, Jul 14 2023

Keywords

Examples

			The triangle begins:
  1;
  0, 2;
  0, 0, 6;
  0, 0, 0, 24;
  0, 0, 0,  0, 120;
  0, 0, 0,  0,   2, 718;
  0, 0, 0,  0,   4,  31, 5005;
  ...
		

Crossrefs

Cf. A000142 (row sums), A350953 (minimal determinant), A350954 (maximal determinant), A351019 (minimal permanent), A351020 (maximal permanent), A356865 (minimal nonzero absolute value determinant), A364231 (right diagonal).

Programs

  • Mathematica
    T[n_,k_]:= Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Range[n]], i]]],{i,n!}],k]; Table[T[n,k],{n,8},{k,n}]//Flatten
  • PARI
    MkMat(v)={matrix(#v, #v, i, j, v[1+abs(i-j)])}
    row(n)={my(f=vector(n)); forperm(vector(n,i,i), v, f[matrank(MkMat(v))]++); f} \\ Andrew Howroyd, Dec 30 2023

Extensions

Terms a(46) and beyond from Andrew Howroyd, Dec 30 2023

A364231 a(n) is the number of n X n nonsingular symmetric Toeplitz matrices using all the integers 1, 2, ..., n.

Original entry on oeis.org

1, 2, 6, 24, 120, 718, 5005, 40274, 362604, 3627915, 39909566, 478991256
Offset: 1

Views

Author

Stefano Spezia, Jul 14 2023

Keywords

Crossrefs

Right diagonal of A364230.
Cf. A350953 (minimal determinant), A350954 (maximal determinant), A351019 (minimal permanent), A351020 (maximal permanent), A356865 (minimal nonzero absolute value determinant).

Programs

  • Mathematica
    a[n_]:=Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Range[n]],i]]],{i,n!}],n]; Array[a,8]

Extensions

a(10)-a(12) from Andrew Howroyd, Dec 30 2023
Showing 1-7 of 7 results.