A355175 Determinant of the n X n matrix [(i-j)^2 + d(i,j)]_{1<=i,j<=n}, where d(i,j) is 1 or 0 according as i = j or not.

1, 1, 0, -9, 45, 931, 6392, 29205, 104497, 315469, 838696, 2018523, 4482765, 9314943, 18301648, 34277321, 61592769, 106738105, 179155504, 292282207, 464869581, 722629755, 1100267400, 1643960605, 2414361521, 3490194501, 4972536856, 6989875075, 9704037421, 13317112215, 18079469856, 24299015697, 32351810305, 42694203377, 55876637664

Offset

0,4

Comments

In 2022, Han Wang and Zhi-Wei Sun determined the values of det[i-j+d(i,j)]_{1<=i,j<=n} and det[|i-j|+d(i,j)]_{1<=i,j<=n}, where d(i,j) is 1 or 0 according as i = j or not.

Links

Table of n, a(n) for n=0..34.

Sela Fried, On Sun's conjectures for determinants of [(i-j)^m + d(i,j)]_{1<=i,j<=n}, 2026.

Sela Fried, Proofs of several OEIS conjectures on determinants and permanents, arXiv:2606.09913 [math.CO], 2026. See pp. 2, 23-25.

Han Wang and Zhi-Wei Sun, Evaluations of three determinants, arXiv:2206.12317 [math.NT], 2022.

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Formula

Conjecture: a(n) = (n^2-4)*(n^2+2*n+3)*(n^5-2*n^4-n^3-28*n^2+60*n-90)/1080.

The conjecture is true (see Fried link). - Sela Fried, Feb 01 2026

G.f.: -(x^9-10*x^8+20*x^7-145*x^6-34*x^5-225*x^4+84*x^3-35*x^2+9*x-1)/(x-1)^10. - Alois P. Heinz, Feb 01 2026

Example

a(3) = -9 since the matrix [(i-j)^2+d(i,j)]_{1<=i,j<=3} = [1,1,4; 1,1,1; 4,1,1] has determinant -9.

Maple

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->`if`(i=j, 1, (i-j)^2))):
seq(a(n), n=0..34);  # Alois P. Heinz, Feb 01 2026

Mathematica

a[n_]:=a[n]=Det[Table[If[i==j, 1, (i-j)^2], {i, 1, n}, {j, 1, n}]]
Table[a[n], {n, 1, 34}]

Prog

(Python)
from sympy import Matrix
def A355175(n): return Matrix(n, n, [(i-j)**2 + int(i==j) for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Jun 28 2022
(PARI) a(n) = matdet(matrix(n, n, i, j, if (i==j, 1, (i-j)^2))); \\ Michel Marcus, Jun 29 2022

Crossrefs

Cf. A000290, A079034.

Sequence in context: A244302 A177742 A322892 * A352398 A261847 A382165

Adjacent sequences: A355172 A355173 A355174 * A355176 A355177 A355178

Keyword

sign,easy

Author

Zhi-Wei Sun, Jun 28 2022

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 01 2026

Status

approved