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.

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A356483 a(n) is the hafnian of a symmetric Toeplitz matrix M(2*n) whose first row consists of prime(1), prime(2), ..., prime(2*n).

Original entry on oeis.org

1, 3, 55, 2999, 347391, 69702479, 22441691645, 10776262328919, 7190279422736061, 6439969796874334809, 7447188585071730451961
Offset: 0

Views

Author

Stefano Spezia, Aug 09 2022

Keywords

Examples

			a(2) = 55 because the hafnian of
    2  3  5  7
    3  2  3  5
    5  3  2  3
    7  5  3  2
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 55.

Links

Crossrefs

Cf. A202038, A336114, A336286, A336400, A338456.
Cf. A356481, A356482, A356484.
Cf. A356490 (determinant of M(n)), A356491 (permanent of M(n)).

Programs

  • Mathematica
    k[i_]:=Prime[i]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, n]], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]
  • PARI
    tm(n) = my(m = matrix(n, n, i, j, if (i==1, prime(j), if (j==1, prime(i))))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

Extensions

a(6) from Michel Marcus, May 02 2023
a(7)-a(10) from Pontus von Brömssen, Oct 14 2023

A356490 a(n) is the determinant of a symmetric Toeplitz matrix M(n) whose first row consists of prime(1), prime(2), ..., prime(n).

Original entry on oeis.org

1, 2, -5, 12, -19, -22, 1143, -4284, 14265, -46726, -84405, 1306096, 32312445, 522174906, 4105967871, 5135940112, -642055973735, -2832632334858, 14310549077571, 380891148658140, 4888186898996125, -49513565563840210, 383405170118692791, -2517836083641473036, -3043377347606882055
Offset: 0

Views

Author

Stefano Spezia, Aug 09 2022

Keywords

Comments

Conjecture: abs(a(n)) is prime only for n = 1, 2, and 4.

Examples

			For n = 1 the matrix M(1) is
    2
with determinant a(1) = 2.
For n = 2 the matrix M(2) is
    2, 3
    3, 2
with determinant a(2) = -5.
For n = 3 the matrix M(3) is
    2, 3, 5
    3, 2, 3
    5, 3, 2
with determinant a(3) = 12.

Links

Crossrefs

Cf. A005843 (trace of M(n)), A309131 (k-superdiagonal sum of M(n)), A356483 (hafnian of M(2*n)), A356491 (permanent of M(n)).
Cf. A350955, A350956.

Programs

  • Maple
    A356490 := proc(n)
    local T,c ;
    if n =0 then
        return 1 ;
    end if;
    T := LinearAlgebra[ToeplitzMatrix]([seq(ithprime(c),c=1..n)],n,symmetric) ;
    LinearAlgebra[Determinant](T) ;
end proc:
seq(A356490(n),n=0..15) ; # R. J. Mathar, Jan 31 2023
  • Mathematica
    k[i_]:=Prime[i]; M[ n_]:=ToeplitzMatrix[Array[k, n]]; a[n_]:=Det[M[n]]; Join[{1},Table[a[n],{n,24}]]
  • PARI
    a(n) = matdet(apply(prime, matrix(n,n,i,j,abs(i-j)+1))); \\ Michel Marcus, Aug 12 2022
  • Python
    from sympy import Matrix, prime
def A356490(n): return Matrix(n,n,[prime(abs(i-j)+1) for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Aug 12 2022

Formula

A350955(n) <= a(n) <= A350956(n).
Showing 1-2 of 2 results.

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