A118709 a(n) = determinant of n X n circulant matrix whose first row is the first n cube numbers 0, 1, ..., (n-1)^3.

1, 0, -1, 513, -532800, 1077540500, -3831689610000, 22051842087895137, -192710430555501494272, 2433436736207275231050384, -42684202683959414242500000000, 1007311823853329619224620155226025, -31149342348518897782279760206406615040, 1234553048404687610083661259955369960321236

Offset

0,4

Links

Table of n, a(n) for n=0..13.

Eric Weisstein's World of Mathematics, Circulant Matrix.

Formula

Contribution from Missouri State University Problem-Solving Group (MSUPSG(AT)MissouriState.edu), May 05 2010: (Start)

a(n) = (-1)^(n-1)*(n - 1)^2*n^(n-2)*(n^(2n) - b(n)^n - c(n)^n + (n^2 - 3*n + 3)^n)/24 for n > 0, where b(n) = (2*n^2 - 3*n - 3 + sqrt(15*n^2 - 18*n - 9)*I)/2, c(n) = (2*n^2 - 3*n - 3 - sqrt(15*n^2 - 18*n - 9)*I)/2, and I the imaginary unit. (End)

Example

a(2) = -1 because of the determinant -1 =
| 0, 1 |
| 1, 0 |.
a(3) = 513 = determinant
|0,1,8|
|8,0,1|
|1,8,0|.
a(6) = 22051842087895137 = determinant
|0,1,8,27,64,125,216|
|216,0,1,8,27,64,125|
|125,216,0,1,8,27,64|
|64,125,216,0,1,8,27|
|27,64,125,216,0,1,8|
|8,27,64,125,216,0,1|
|1,8,27,64,125,216,0|.

Maple

a:= n-> LinearAlgebra[Determinant](Matrix(n, shape=Circulant[[i^3$i=0..n-1]])):
seq(a(n), n=0..14);  # Alois P. Heinz, Feb 21 2026

Mathematica

Join[{1}, Table[Det[Table[RotateRight[Range[0, i-1]^3, n], {n, 0, i-1}]], {i, 13}]] (* Harvey P. Dale, Oct 22 2012 *)
b[n_]:=(2*n^2-3*n-3+Sqrt[15n^2-18n-9]I)/2; c[n_]:=(2*n^2-3*n-3-Sqrt[15n^2-18n-9]I)/2; a[0]=1; a[n_]:=(-1)^(n-1)*(n-1)^2*n^(n-2)*(n^(2n)-b[n]^n-c[n]^n+(n^2-3n+3)^n)/24; Simplify[Array[a, 14, 0]] (* Stefano Spezia, Feb 21 2026 *)

Crossrefs

See also: A000578 The cubes: a(n) = n^3. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000578, A048954, A052182, A066933, A086459, A086569.

Sequence in context: A094647 A236431 A384935 * A392679 A296145 A283369

Adjacent sequences: A118706 A118707 A118708 * A118710 A118711 A118712

Keyword

easy,sign

Author

Jonathan Vos Post, May 20 2006

Extensions

More terms from Harvey P. Dale, Oct 22 2012

a(0)=1 prepended by Alois P. Heinz, Feb 21 2026

Status

approved