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 1-8 of 8 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

Links

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

A358323 a(n) is the minimal determinant of an n X n symmetric Toeplitz matrix using the integers 0 to n - 1.

Original entry on oeis.org

1, 0, -1, -7, -60, -1210, -34020, -607332, -30448441, -1093612784, -55400732937, -2471079070511, -197500419383964
Offset: 0

Views

Author

Stefano Spezia, Nov 09 2022

Keywords

Examples

			a(3) = -7:
    [1, 2, 0;
     2, 1, 2;
     0, 2, 1]
a(4) = -60:
    [2, 3, 0, 1;
     3, 2, 3, 0;
     0, 3, 2, 3;
     1, 0, 3, 2]
a(5) = -1210:
    [4, 3, 0, 2, 1;
     3, 4, 3, 0, 2;
     0, 3, 4, 3, 0;
     2, 0, 3, 4, 3;
     1, 2, 0, 3, 4]

Links

Crossrefs

Cf. A350953.
Cf. A358324 (maximal), A358325 (minimal nonzero absolute value), A358326 (minimal permanent), A358327 (maximal permanent).

Programs

  • Mathematica
    Join[{1}, Table[Min[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

A359614 a(n) is the minimal 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, -30, -256, -7595, -358301, -7665804, -227965955, -13089461984, -2467071630448
Offset: 0

Views

Author

Stefano Spezia, Jan 07 2023

Keywords

Examples

			a(4) = -256:
  [   4,  3*i,  2*i,   i;
   -3*i,    4,  3*i, 2*i;
   -2*i, -3*i,    4, 3*i;
     -i, -2*i, -3*i,   4 ]

Links

Crossrefs

Cf. A350953, A359559, A359561.
Cf. A359615 (maximal), A359616 (minimal permanent), A359617 (maximal permanent).

Programs

  • Mathematica
    a={1}; For[n=1, n<=8, n++, mn=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]]]])
  • Python
    from itertools import permutations
from sympy import Matrix, I
def A359614(n): return min(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)

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).

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]

Links

Crossrefs

Cf. A350953, A374139.
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

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 1-8 of 8 results.

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

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