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 1-5 of 5 results.

A318173 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n) and whose first column consists of prime(1), prime(n + 1), ..., prime(2*n - 1).

Original entry on oeis.org

2, -11, 158, -6513, 202790, -12710761, 578257422, -45608219247, 8774909485920, -579515898830751, 115918088707226940, -16737522590543449641, 1282860173728469083872, -189053227741259934603831, 55171097827950314187327460, -16235234399834578732807710581
Offset: 1

Views

Author

Stefano Spezia, Aug 20 2018

Keywords

Comments

The trace of the matrix M(n) is A005843(n).
The sum of the first row of the matrix M(n) is A007504(n).
The permanent of the matrix M(n) is A306457(n).
For n > 1, the subdiagonal sum of the matrix M(n) is A306192(n).

Examples

			For n = 1 the matrix M(1) is
   2
with determinant Det(M(1)) = 2.
For n = 2 the matrix M(2) is
   2, 3
   5, 2
with Det(M(2)) = -11.
For n = 3 the matrix M(3) is
   2, 3, 5
   7, 2, 3
  11, 7, 2
with Det(M(3)) = 158.

Links

Crossrefs

Cf. A005843, A000040, A007504, A306457, A306192.

Programs

  • Maple
    f:= proc(n) uses LinearAlgebra;
Determinant(ToeplitzMatrix([seq(ithprime(i),i=2*n-1..n+1,-1),seq(ithprime(i),i=1..n)]))
end proc:
map(f, [$1..20]); # Robert Israel, Aug 30 2018
  • Mathematica
    p[i_]:=Prime[i]; a[n_]:=Det[ToeplitzMatrix[Join[{p[1]},Array[p,n-1,{n+1,2*n-1}]],Array[p,n]]]; Array[a,20]
  • PARI
    tm(n) = {my(m = matrix(n, n, i, j, if (i==1, prime(j), if (j==1, prime(n+i-1))))); for (i=2, n, for (j=2, n, m[i,j] = m[i-1, j-1];);); m;}
a(n) = matdet(tm(n)); \\ Michel Marcus, Mar 17 2019

A374067 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th prime or 1 if i = j.

Original entry on oeis.org

1, 1, 5, 42, 753, 22969, 1226225, 98413280, 11551199289, 1828335971613, 379823112871605, 102232301626742202, 34359550765856135217, 14289766516805617273497, 7224166042347461997365713, 4334493536305030883929928032, 3046742350470292308074313518937, 2492781304663024301187012794633153
Offset: 0

Views

Author

Stefano Spezia, Jun 27 2024

Keywords

Comments

Conjecture: a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the first n-1 primes off-diagonal. - Stefano Spezia, Jul 08 2024

Examples

			a(4) = 753:
  [1, 2, 3, 5]
  [2, 1, 2, 3]
  [3, 2, 1, 2]
  [5, 3, 2, 1]

Crossrefs

Cf. A071078 (determinant), A306457, A318173.
Cf. A374068, A374069, A374070, A374071.

Programs

  • Mathematica
    a[n_]:=Permanent[Table[ If[i == j, 1, Prime[Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Join[{1},Array[a, 17]]
  • PARI
    a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 1, prime(abs(i-j))))); \\ Michel Marcus, Jun 27 2024

A374068 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals the |i-j|-th prime or 0 if i = j.

Original entry on oeis.org

1, 0, 4, 24, 529, 16100, 919037, 75568846, 9196890092, 1491628025318, 317579623173729, 86997150829931700, 29703399282858184713, 12512837775355494800500, 6397110844644502402189404, 3875565057688532269985283868, 2747710211567246171588232074225, 2265312860218073375019946448731300
Offset: 0

Views

Author

Stefano Spezia, Jun 27 2024

Keywords

Comments

Conjecture: a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the first n-1 primes off-diagonal. - Stefano Spezia, Jul 06 2024

Examples

			a(4) = 529:
  [0, 2, 3, 5]
  [2, 0, 2, 3]
  [3, 2, 0, 2]
  [5, 3, 2, 0]

Crossrefs

Cf. A071079 (determinant), A085807, A306457, A318173.
Cf. A374067, A374069, A374070, A374071.

Programs

  • Mathematica
    a[n_]:=Permanent[Table[If[i == j, 0, Prime[Abs[i - j]]], {i, 1, n}, {j, 1, n}]]; Join[{1},Array[a, 17]]
  • PARI
    a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 0, prime(abs(i-j))))); \\ Michel Marcus, Jun 28 2024

A322909 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n and whose first column consists of 1, n + 1, ..., 2*n - 1.

Original entry on oeis.org

1, 1, 7, 100, 2840, 129428, 8613997, 791557152, 95921167710, 14818153059968, 2842735387366627, 663020104070865664, 184757202542187563476, 60623405966739216871680, 23135486197103263598936745, 10160292704659539620791062528, 5087671168376607498331875818106
Offset: 0

Views

Author

Stefano Spezia, Dec 30 2018

Keywords

Comments

The matrix M(n) differs from that of A306457 in using successive positive integers in place of successive prime numbers. [Modified by Stefano Spezia, Dec 20 2019 at the suggestion of Michel Marcus]
The trace of M(n) is A000027(n).
The sum of the first row of M(n) is A000217(n).
The sum of the first column of M(n) is A005448(n). [Corrected by Stefano Spezia, Dec 19 2019]
For n > 1, the sum of the superdiagonal of M(n) is A005843(n).
For n > 0, the sum of the (k-1)-th superdiagonal of M(n) is A003991(n,k). - Stefano Spezia, Dec 29 2019
For n > 1 and k > 0, the sum of the k-th subdiagonal of M(n) is A120070(n,k). - Stefano Spezia, Dec 31 2019

Examples

			For n = 1 the matrix M(1) is
   1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
   1, 2
   3, 1
with permanent a(2) = 7.
For n = 3 the matrix M(3) is
   1, 2, 3
   4, 1, 2
   5, 4, 1
with permanent a(3) = 100.

Links

Crossrefs

Cf. A000027, A000217, A003991, A005448, A005843, A120070, A306457, A322908 (determinant of M(n)).

Programs

  • Maple
    with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(ToeplitzMatrix([
         seq(i, i=2*n-1..n+1, -1), seq(i, i=1..n)]))):
seq(a(n), n = 0 .. 15);
  • Mathematica
    b[n_]:=n; a[n_]:=If[n==0,1,Permanent[ToeplitzMatrix[Join[{b[1]}, Array[b, n-1, {n+1, 2*n-1}]], Array[b, n]]]]; Array[a, 15,0]
  • PARI
    tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j, if (j==1, n+i-1)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
a(n) = matpermanent(tm(n)); \\ Stefano Spezia, Dec 19 2019

Extensions

a(0) = 1 prepended by Stefano Spezia, Dec 19 2019

A381514 a(n) is the hafnian of a symmetric Toeplitz matrix of order 2*n whose off-diagonal element (i,j) equals the |i-j|-th prime.

Original entry on oeis.org

1, 2, 23, 899, 85072, 15120411, 4439935299, 1989537541918, 1264044973158281, 1090056235155152713, 1227540523199054294506
Offset: 0

Views

Author

Stefano Spezia, Feb 25 2025

Keywords

Examples

			a(2) = 23 because the hafnian of
  [d  2  3  5]
  [2  d  2  3]
  [3  2  d  2]
  [5  3  2  d]
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 2*2 + 3*3 + 5*2 = 23. Here d denotes the generic element on the main diagonal of the matrix from which the hafnian does not depend.

Links

Crossrefs

Cf. A374067, A374068.
Cf. A071078, A071079, A085807, A306457, A318173, A356483.

Programs

  • Mathematica
    M[i_, j_]:=Prime[Abs[i-j]]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i]], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 5, 0]

Extensions

a(5)-a(10) from Pontus von Brömssen, Feb 26 2025
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