1,2

det(C(n))=0 for n divisible by 6.

The determinant of the circulant matrix is 0 when 6 divides n because the polynomial (x+1)^(6k) - 1 has roots that are roots of unity. See A086569 for a generalization. - T. D. Noe (noe(AT)sspectra.com), Jul 21 2003

P. Ribenboim, "Fermat's Last Theorem for Amateurs", Springer-Verlag, NY, 1999, pp. 126, 136.

Anastasios Simalarides, "Upper bounds for the prime divisors of Wendt's determinant", Math. Comp., 71(2002),415-427.

P. Ribenboim, 13 Lectures on Fermat's last theorem, Springer-Verlag, NY, 1979, pp. 61-63. MR0551363 (81f:10023)

T. D. Noe, Table of n, a(n) for n=1..50

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114) [From Gerard P. Michon (g.michon(AT)att.net), Jan 16 2009]

a(n)=0 if and only if 6 divides n. If d divides n, then a(d) divides a(n). - Michael Somos Apr 03 2007

a[n_] := Resultant[x^n-1, (1+x)^n-1, x]

(PARI) a(n)=if(n<1, 0, matdet(matrix(n, n, i, j, binomial(n, (j-i)%n))))

(PARI) {a(n)= if(n<1, 0, matdet( matrix( n, n, i, j, binomial( n, (j-i)%n ))))}

Cf. A052182 (circulant of natural numbers), A066933 (circulant of prime numbers), A086459 (circulant of powers of 2), A086569.

See A096964 for another definition.

A129205(n)^2*(1-4^n) = a(2*n).

Sequence in context: A072343 A151423 A161605 this_sequence A086569 A143636 A060545

Adjacent sequences: A048951 A048952 A048953 this_sequence A048955 A048956 A048957

sign,nice

Eric Weisstein (eric(AT)weisstein.com)

Additional comments from Michael Somos, May 27 2000 and Dec 16 2001