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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 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 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. 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, 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 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 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 ))))} 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 A264639 A143636 Adjacent sequences:  A048951 A048952 A048953 * A048955 A048956 A048957 KEYWORD sign,nice AUTHOR EXTENSIONS Additional comments from Michael Somos, May 27 2000 and Dec 16 2001 STATUS approved

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