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 1-9 of 9 results.

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

Links

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

A359615 a(n) is the maximal determinant of an n X n Hermitian Toeplitz matrix using all the integers 1, 2, ..., n and with all off-diagonal elements purely imaginary.

Original entry on oeis.org

1, 1, 3, 9, 512, 9195, 242931, 7459494, 524426191, 17012915860, 773407040859
Offset: 0

Views

Author

Stefano Spezia, Jan 07 2023

Keywords

Examples

			a(4) = 512:
  [   1,  4*i,  2*i, 3*i;
   -4*i,    1,  4*i, 2*i;
   -2*i, -4*i,    1, 4*i;
   -3*i, -2*i, -4*i,   1 ]

Links

Crossrefs

Cf. A350954, A359559, A359561.
Cf. A359614 (minimal), A359616 (minimal permanent), A359617 (maximal permanent).

Programs

  • Mathematica
    a={1}; For[n=1, n<=8, n++, mx=-Infinity; For[d=1, d<=n, d++, For[i=1, i<=(n-1)!, i++, If[(t=Det[ToeplitzMatrix[Join[{d}, I Part[Permutations[Drop[Range[n], {d}]], i]]]])>mx, mx=t]]]; AppendTo[a, mx]]; a
  • Python
    from itertools import permutations
from sympy import Matrix, I
def A359615(n): return max(Matrix(n,n,[(d[i-j] if i>j else -d[j-i]) if i!=j else d[0]*I for i in range(n) for j in range(n)]).det()*(1,-I,-1,I)[n&3] for d in permutations(range(1,n+1))) # Chai Wah Wu, Jan 25 2023

A356865 Minimal absolute value of determinant of a nonsingular n X n symmetric Toeplitz matrix using the integers 1 to n.

Original entry on oeis.org

1, 1, 3, 8, 12, 3, 13, 19, 5, 5, 1, 3, 1
Offset: 0

Views

Author

Lucas A. Brown, Sep 01 2022

Keywords

Examples

			a(3) = 8:
    1  2  3
    2  1  2
    3  2  1
a(4) = 12:
    2  1  3  4
    1  2  1  3
    3  1  2  1
    4  3  1  2
a(5) = 3:
    1  5  2  3  4
    5  1  5  2  3
    2  5  1  5  2
    3  2  5  1  5
    4  3  2  5  1

Links

Crossrefs

Cf. A350953 (minimal signed), A350954 (maximal signed), A348891 (prime entries).

Programs

  • Python
    from itertools import permutations
from sympy import Matrix
def A348891(n): return min(d for d in (abs(Matrix([p[i:0:-1]+p[0:n-i] for i in range(n)]).det()) for p in permutations(range(1,n+1))) if d > 0)

A358324 a(n) is the maximal determinant of an n X n symmetric Toeplitz matrix using the integers 0 to n - 1.

Original entry on oeis.org

1, 0, 1, 8, 63, 2090, 36875, 1123653, 34292912, 1246207300, 53002204560, 2418538080316, 215120941720912
Offset: 0

Views

Author

Stefano Spezia, Nov 09 2022

Keywords

Examples

			a(3) = 8:
    [0, 2, 1;
     2, 0, 2;
     1, 2, 0]
a(4) = 63:
    [1, 3, 2, 0;
     3, 1, 3, 2;
     2, 3, 1, 3;
     0, 2, 3, 1]
a(5) = 2090:
    [2, 4, 0, 1, 3;
     4, 2, 4, 0, 1;
     0, 4, 2, 4, 0;
     1, 0, 4, 2, 4;
     3, 1, 0, 4, 2]

Links

Crossrefs

Cf. A350954.
Cf. A358323 (minimal), A358325 (minimal nonzero absolute value), A358326 (minimal permanent), A358327 (maximal permanent).

Programs

  • Mathematica
    Join[{1}, Table[Max[Table[Det[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n-1]]], i]]],{i,n!}]],{n,9}]]

Extensions

a(10)-a(12) from Lucas A. Brown, Nov 16 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

Links

Crossrefs

Cf. A000142, A350953, A350954, A356865.
Cf. A369831, A369832, A369833, A369834, A369835.

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).

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]

Links

Crossrefs

Cf. A350954, A374139.
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

A351609 Maximal absolute value of the determinant of an n X n symmetric matrix using the integers 1 to n*(n + 1)/2.

Original entry on oeis.org

1, 1, 7, 152, 7113, 745285, 94974369
Offset: 0

Views

Author

Stefano Spezia, Feb 14 2022

Keywords

Comments

Upper bounds for the next terms can be found by considering all possibilities of choosing matrix entries on the diagonal and applying Gasper's determinant theorem (see references in A085000): a(7) <= 22475584128, a(8) <= 6634478203404, a(9) <= 2647044512044258. - Hugo Pfoertner, Feb 18 2022

Examples

			a(3) = 152:
   2    4    6
   4    5    1
   6    1    3
a(4) = 7113:
   2    6    8    9
   6    5   10    1
   8   10    3    4
   9    1    4    7

Crossrefs

Cf. A000217, A351147, A351148, A351153.
Cf. A085000, A180128, A350931, A350933, A350954, A350956.

Formula

a(n) = max(abs(A351147(n)), A351148(n)). - Hugo Pfoertner, Feb 16 2022

Extensions

a(5)-a(6) from Hugo Pfoertner, Feb 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;
  ...

Links

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

Links

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 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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