OFFSET
1,2
COMMENTS
The n-th principal submatrix of A143182 is an n X n symmetric Toeplitz matrix whose first row consists of successive natural numbers 1, ..., n. - Stefano Spezia, Sep 23 2018
Conjecture: a(1) and a(2) are the only terms that are odd numbers. - Stefano Spezia, Oct 28 2018
The conjecture is true. This is because the permanent and determinant of every integer matrix are congruent modulo 2. The determinant of the matrix here is known to be (-1)^(n-1)*(n+1)*2^(n-2), i.e., even for n >= 3 (see Cloitre's comment in A001792). - Sela Fried, Feb 11 2026
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..35 (terms 1..14 from Clark Kimberling, terms 15..22 from Stefano Spezia)
MAPLE
f:= proc(n) uses LinearAlgebra;
Permanent(ToeplitzMatrix([seq(i, i=1 ..n)], n, symmetric))
end proc:
map(f, [$1..20]); # Stefano Spezia, Oct 28 2018
MATHEMATICA
f[i_, j_] := Max[i - j + 1, j - i + 1]; (* A143182 *)
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[4]] (* 4 X 4 principal submatrix *)
Table[Det[m[n]], {n, 1, 22}] (* A001792 - signed *)
Permanent[m_] :=
With[{a = Array[x, Length[m]]},
Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 14}] (* A204235 *)
b[i_]:=i; a[n_]:=Permanent[ToeplitzMatrix[Array[b, n], Array[b, n]]]; Array[a, 22] (* Stefano Spezia, Sep 23 2018 *)
PROG
(PARI) {a(n) = matpermanent(matrix(n, n, i, j, max(i - j + 1, j - i + 1)))}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Apr 29 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jan 13 2012
EXTENSIONS
Extended by Stefano Spezia, Oct 28 2018
STATUS
approved
