OFFSET
0,4
COMMENTS
Conjecture: For any odd prime p, we have a(p) == -1/2 (mod p). - Zhi-Wei Sun, Aug 30 2021
Conjecture: a(n) is the minimal 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. - Stefano Spezia, Jul 05 2024
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..37
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.
MAPLE
with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> abs(i-j)))):
seq(a(n), n=0..18); # Alois P. Heinz, Nov 14 2016
MATHEMATICA
a[n_]:=Permanent[Table[Abs[i - j], {i, n}, {j, n}]]; Join[{1}, Array[a, 18]] (* Stefano Spezia, Jun 28 2024 *)
PROG
(PARI) permRWNb(a)= n=matsize(a)[1]; if(n==1, return(a[1, 1])); sg=1; in=vectorv(n); x=in; x=a[, n]-sum(j=1, n, a[, j])/2; p=prod(i=1, n, x[i]); for(k=1, 2^(n-1)-1, sg=-sg; j=valuation(k, 2)+1; z=1-2*in[j]; in[j]+=z; x+=z*a[, j]; p+=prod(i=1, n, x[i], sg)); return(2*(2*(n%2)-1)*p)
for(n=1, 22, a=matrix(n, n, i, j, abs(i-j)); print1(permRWNb(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007
(PARI) {a(n) = matpermanent(matrix(n, n, i, j, abs(i-j)))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 12 2021
(Python)
from sympy import Matrix
def A085807(n): return Matrix(n, n, [abs(j-k) for j in range(n) for k in range(n)]).per() # Chai Wah Wu, Sep 14 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 24 2003
EXTENSIONS
More terms from Vladeta Jovovic, Jul 26 2003
a(0)=1 prepended by Alois P. Heinz, Nov 14 2016
STATUS
approved