cp's OEIS Frontend

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.

%I A303260 #19 Oct 18 2021 18:29:23
%S A303260 1,1,4,28,273,3421,52288,941578,19505545,456790123,11931215316,
%T A303260 343871642632,10840081272265,371026432467913,13702802011918048,
%U A303260 543154131059225686,23000016472483168305,1036227971225610466711,49492629462587441963140,2497992686980609418282548,132849300060919364474261281
%N 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.
%C A303260 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.
%C A303260 a(0) = 1 is (by convention) the determinant of a 0 X 0 matrix.
%H A303260 Max Alekseyev, <a href="/A219324/a219324.png">Illustration for a(9) = 456790123 = A219324(20)</a>.
%H A303260 N. I. Belukhov, <a href="http://www.mccme.ru/free-books/matpros/mpf.pdf">Solution to Problem 14.7</a> (in Russian), Matematicheskoe Prosveshchenie 15 (2011), pp. 241-244.
%H A303260 Wikipedia, <a href="http://en.wikipedia.org/wiki/Circulant_matrix">Circulant matrix</a>.
%F A303260 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.
%e A303260 a(5) = 3421 is the determinant of the matrix
%e A303260    ( 5 0 1 2 3 )
%e A303260    ( 3 5 0 1 2 )
%e A303260    ( 2 3 5 0 1 )  and 3421 = 23501[6], i.e., written in base 6.
%e A303260    ( 1 2 3 5 0 )
%e A303260    ( 0 1 2 3 5 ).
%o A303260 (PARI) a(n)=matdet(matrix(n,n,i,j,(j-i-1)%n+(i==j)))
%o A303260 (Python)
%o A303260 from sympy import Matrix
%o A303260 def A303260(n): return Matrix(n,n, lambda i,j:(j-i-1) % n + (i==j)).det() # _Chai Wah Wu_, Oct 18 2021
%Y A303260 Cf. A081131(n+1) = determinant of the circulant matrix C(n) defined in formula, A070896 (signed variant).
%Y A303260 See also A219324.
%K A303260 nonn
%O A303260 0,3
%A A303260 _M. F. Hasler_, Apr 23 2018