A348183 a(n) is the determinant of the n X n matrix M = (m_{i,j}), i,j from 0 to n-1: m{i,j} = (i+j)^2 mod n.

1, 0, -1, -2, 0, 250, 5616, -33614, 0, 204073344, -900000000, -9431790764, 0, 10752962364222, -1870899108384768, -36328974609375000, 0, 22899384412078526344, -111400529859275793629184, -43843094862278417487512, 0, 2870507605405055660542502550, 67015802375208384199755038720

Offset: 0

Author

Dennis Gruhlke and Albert Böschow, Oct 05 2021

Keywords

• sign

Comments

It seems that for values of n divisible by 4 -> a(n) = 0 and rank(M) = n/2.

Examples

			a(2) =  |0^2 mod 2, 1^2 mod 2| = -1
        |1^2 mod 2, 2^2 mod 2|
--
        |0^2 mod 3, 1^2 mod 3, 2^2 mod 3|
a(3) =  |1^2 mod 3, 2^2 mod 3, 3^2 mod 3| = -2
        |2^2 mod 3, 3^2 mod 3, 4^2 mod 3|
		

Crossrefs

Cf. A070896.

Programs

Mathematica

a[n_]:=Table[Mod[(i+j)^2,n],{i,0,n-1},{j,0,n-1}]; Join[{1},Table[Det[a[n]], {n, 22}]] (* Stefano Spezia, Oct 06 2021 *)

PARI

a(n) = matdet(matrix(n, n, i, j, i--; j--; (i+j)^2 % n)); \\ Michel Marcus, Oct 06 2021

Python

from sympy import Matrix
def A348183(n): return Matrix(n,n,[pow(i+j,2,n) for i in range(n) for j in range(n)]).det() # Chai Wah Wu, Nov 24 2021

Extensions

a(14)-a(17) corrected by and more terms from Stefano Spezia, Oct 06 2021.