A303260 Determinant of n X n matrix A[i,j] = (j - i - 1 mod n) + [i=j], i.e., the circulant having (n, 0, 1, ..., n-2) as first row.

1, 1, 4, 28, 273, 3421, 52288, 941578, 19505545, 456790123, 11931215316, 343871642632, 10840081272265, 371026432467913, 13702802011918048, 543154131059225686, 23000016472483168305, 1036227971225610466711, 49492629462587441963140, 2497992686980609418282548, 132849300060919364474261281

Offset

0,3

Comments

It is remarkable that for odd n, this determinant has its base n+1 digits equal to the middle row: e.g., a(9) = 456790123 is the determinant of the circulant matrix having [4,5,6,7,9,0,1,2,3] as middle row.

a(0) = 1 is (by convention) the determinant of a 0 X 0 matrix.

Links

Table of n, a(n) for n=0..20.

Max Alekseyev, Illustration for a(9) = 456790123 = A219324(20).

N. I. Belukhov, Solution to Problem 14.7 (in Russian), Matematicheskoe Prosveshchenie 15 (2011), pp. 241-244.

Wikipedia, Circulant matrix.

Formula

a(n) = det(I(n) + C(n)), where I(n) is the n X n identity matrix and C(n) is the circulant having (n-1, ..., 0) as first column.

Example

a(5) = 3421 is the determinant of the matrix
   ( 5 0 1 2 3 )
   ( 3 5 0 1 2 )
   ( 2 3 5 0 1 )  and 3421 = 23501[6], i.e., written in base 6.
   ( 1 2 3 5 0 )
   ( 0 1 2 3 5 ).

Prog

(PARI) a(n)=matdet(matrix(n, n, i, j, (j-i-1)%n+(i==j)))
(Python)
from sympy import Matrix
def A303260(n): return Matrix(n, n, lambda i, j:(j-i-1) % n + (i==j)).det() # Chai Wah Wu, Oct 18 2021

Crossrefs

Cf. A081131(n+1) = determinant of the circulant matrix C(n) defined in formula, A070896 (signed variant).

See also A219324.

Sequence in context: A081917 A302583 A302605 * A174494 A128318 A032274

Adjacent sequences: A303257 A303258 A303259 * A303261 A303262 A303263

Keyword

nonn

Author

M. F. Hasler, Apr 23 2018

Status

approved