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.

A369832 a(n) is the number of distinct values of the determinant of an n X n symmetric Toeplitz matrix using the first n prime numbers.

Original entry on oeis.org

1, 1, 2, 6, 24, 116, 717, 5033, 40301, 362845
Offset: 0

Views

Author

Stefano Spezia, Feb 03 2024

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := CountDistinct[Table[Det[ToeplitzMatrix[Part[Permutations[Prime[Range[n]]], i]]], {i, n !}]]; Join[{1}, Array[a,9]]
  • Python
    from itertools import permutations
    from sympy import primerange, prime, Matrix
    def A369832(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).det() for p in permutations(primerange(prime(n)+1))}) if n else 1 # Chai Wah Wu, Feb 11 2024

Formula

a(n) <= A000142(n).

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).

A369834 a(n) is the number of distinct values of the determinant of an n X n symmetric Toeplitz matrix using the integers 0 to n-1.

Original entry on oeis.org

1, 1, 2, 5, 23, 94, 614, 4628, 38243, 351024
Offset: 0

Views

Author

Stefano Spezia, Feb 03 2024

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := CountDistinct[Table[Det[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n - 1]]], i]]], {i, n !}]]; Join[{1}, Array[a,9]]
  • Python
    from itertools import permutations
    from sympy import Matrix
    def A369834(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).det() for p in permutations(range(n))}) # Chai Wah Wu, Feb 11 2024

Formula

a(n) <= A000142(n).

A369831 a(n) is the number of distinct values of the permanent of an n X n symmetric Toeplitz matrix using the integers 1 to n.

Original entry on oeis.org

1, 1, 1, 6, 23, 120, 720, 5040, 40320, 362880
Offset: 0

Views

Author

Stefano Spezia, Feb 03 2024

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := CountDistinct[Table[Permanent[ToeplitzMatrix[Part[Permutations[Range[n]],i]]], {i, n!}]]; Join[{1}, Array[a,9]]
  • Python
    from itertools import permutations
    from sympy import Matrix
    def A369831(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).per() for p in permutations(range(1,n+1))}) # Chai Wah Wu, Feb 12 2024

Formula

a(n) <= A000142(n).
Conjectured e.g.f.: 1/(1 - x) - x^2/2 - x^4/24.

A369835 a(n) is the number of distinct values of the permanent of an n X n symmetric Toeplitz matrix using the integers 0 to n-1.

Original entry on oeis.org

1, 1, 1, 6, 23, 119, 718, 5038, 40320, 362879
Offset: 0

Views

Author

Stefano Spezia, Feb 03 2024

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := CountDistinct[Table[Permanent[ToeplitzMatrix[Part[Permutations[Join[{0}, Range[n - 1]]], i]]], {i, n !}]]; Join[{1}, Array[a,9]]
  • Python
    from itertools import permutations
    from sympy import Matrix
    def A369835(n): return len({Matrix([p[i:0:-1]+p[:n-i] for i in range(n)]).per() for p in permutations(range(n))}) # Chai Wah Wu, Feb 11 2024

Formula

a(n) <= A000142(n).

A374621 Expansion of e.g.f. 1 - x^4/24 - log(1 - x).

Original entry on oeis.org

1, 1, 1, 2, 5, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000, 25852016738884976640000
Offset: 0

Views

Author

Stefano Spezia, Jul 14 2024

Keywords

Comments

Conjectures (confirmed up to n = 10): (Start)
a(n) is the number of distinct values of the permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.
a(n) is the number of distinct values of the permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the first n-1 primes off-diagonal. (End)

Crossrefs

Programs

  • Mathematica
    nmax=24; CoefficientList[Series[1-x^4/24-Log[1-x], {x,0,nmax}], x]*Range[0,nmax]!

Formula

a(0) = 1, a(4) = 5, and a(n) = (n-1)! for n > 0 and n <> 4.
a(n) = (n - 1)*a(n-1) for n > 5.
Showing 1-6 of 6 results.