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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. 4
1, 1, 4, 28, 273, 3421, 52288, 941578, 19505545, 456790123, 11931215316, 343871642632, 10840081272265, 371026432467913, 13702802011918048, 543154131059225686, 23000016472483168305, 1036227971225610466711, 49492629462587441963140, 2497992686980609418282548, 132849300060919364474261281 (list; graph; refs; listen; history; text; internal format)
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)))

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

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Last modified December 14 09:41 EST 2019. Contains 329979 sequences. (Running on oeis4.)