A048954 Wendt determinant of n-th circulant matrix C(n).

1, -3, 28, -375, 3751, 0, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, 0, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368, -1562716604740038367719196682456673375

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

C(n) is the matrix whose first row is [c_1, ..., c_n] where c_i = binomial(n,i-1), and subsequent rows are obtained by cyclically shifting the previous row one place to the right: see examples and PARI code. - M. F. Hasler, Dec 17 2016

References

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 W. Boyd, The asymptotic behaviour of the binomial circulant determinant, Journal of Mathematical Analysis and Applications, Volume 86, Issue 1, March 1982, Pages 30-38.

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

D. Burde and W. A. Moens, The structure of Lie algebras with a derivation satisfying a polynomial identity, arXiv:2009.05434 [math.RA], 2020.

L. Carlitz, A determinant connected with Fermat's last theorem, Proc. Amer. Math. Soc. 10 (1959), 686-690.

L. Carlitz, A determinant connected with Fermat's last theorem, Proc. Amer. Math. Soc. 11 (1960), 730-733.

Joshua Cooper and Zhibin Du, Note on the spectra of Steiner distance hypermatrices, arXiv:2403.02287 [math.CO], 2024. See pp. 2, 4.

Greg Fee and Andrew Granville, The prime factors of Wendt's binomial circulant determinant, Math. Comp. 57 (1991), 839-848.

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

J. S. Frame, Factors of the binomial circulant determinant, Fibonacci Quart., 18 (1980), pp. 9-23.

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

Charles Helou and Guy Terjanian, Arithmetical properties of wendt's determinant, Journal of Number Theory, Volume 115, Issue 1, November 2005, Pages 45-57.

Emma Lehmer, On a resultant connected with Fermat's last theorem, Bull. Amer. Math. Soc. 41 (1935), 864-867.

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

E. Wendt, Arithmetische Studien über den letzten Fermatschen Satz, welcher aussagt, dass die Gleichung a^n=b^n+c^n für n>2 in ganzen Zahlen nicht auflösbar ist, Reimer (Berlin), 1894.

Formula

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

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

Example

a(2) = det [ 1 2 ; 2 1 ] = -3.
a(3) = det [ 1 3 3 ; 3 1 3 ; 3 3 1 ] = 28.
a(4) = det [ 1 4 6 4 ; 4 1 4 6 ; 6 4 1 4 ; 4 6 4 1 ] = -375.

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) = polresultant( x^n - 1, (1+x)^n - 1, x )

Crossrefs

Cf. A052182 (circulant of natural numbers), A066933 (circulant of prime numbers), A086459 (circulant of powers of 2), A086569, A129205, A215615, A215616.

See A096964 for another definition.

Sequence in context: A352383 A151423 A161605 * A086569 A264639 A298696

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