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A048954 Wendt determinant of n-th circulant matrix C(n). 16
1, -3, 28, -375, 3751, 0, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, 0, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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, Jul 21 2003

E. Lehmer claimed, and J. S. Frame proved, that 2^n - 1 divides  a(n) and the quotient abs(a(n))/(2^n - 1) is a perfect square (Ribenboim 1999, p. 128). - Jonathan Sondow, Aug 17 2012

REFERENCES

E. Brown and M. Chamberland, Generalizing Gauss's Gem, Amer. Math. Monthly, 119 (No. 7, 2012), 597-601. - N. J. A. Sloane, Sep 07 2012

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

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

LINKS

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

David Ford and Vijay Jha, On Wendt’s Determinant and Sophie Germain’s Theorem, Experimental Mathematics, 2 (1993) No. 2, 113-120.

Charles Helou, On Wendt's Determinant, Math. Comp., 66 (1997) No. 219, 1341-1346.

Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114)

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

Eric Weisstein's World of Mathematics, Circulant matrix

FORMULA

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) = (-1)^(n-1) * (2^n - 1) * A215615(n)^2. - Jonathan Sondow, Aug 17 2012

a(2*n) = -3*A215616(n)^3. - Jonathan Sondow, Aug 18 2012

MATHEMATICA

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

PROG

(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 ))))}

CROSSREFS

Cf. A052182 (circulant of natural numbers), A066933 (circulant of prime numbers), A086459 (circulant of powers of 2), A086569, A215615, A215616.

See A096964 for another definition.

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

Sequence in context: A212032 A151423 A161605 * A086569 A143636 A219532

Adjacent sequences:  A048951 A048952 A048953 * A048955 A048956 A048957

KEYWORD

sign,nice

AUTHOR

Eric W. Weisstein

EXTENSIONS

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

STATUS

approved

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Last modified November 28 21:04 EST 2014. Contains 250406 sequences.