

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, ..., n2) as first row.


5



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



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 (n1, ..., 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, (ji1)%n+(i==j)))
(Python)
from sympy import Matrix
def A303260(n): return Matrix(n, n, lambda i, j:(ji1) % 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).


KEYWORD

nonn


AUTHOR



STATUS

approved



