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%I #63 Mar 09 2024 10:54:50
%S 1,-3,28,-375,3751,0,6835648,-1343091375,364668913756,
%T -210736858987743,101832157445630503,0,487627751563388801409591,
%U -4875797582053878382039400448,58623274842128064372315087290368,-1562716604740038367719196682456673375
%N Wendt determinant of n-th circulant matrix C(n).
%C det(C(n)) = 0 for n divisible by 6.
%C 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
%C 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 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
%D P. Ribenboim, "Fermat's Last Theorem for Amateurs", Springer-Verlag, NY, 1999, pp. 126, 136.
%D P. Ribenboim, 13 Lectures on Fermat's last theorem, Springer-Verlag, NY, 1979, pp. 61-63. MR0551363 (81f:10023).
%H T. D. Noe, <a href="/A048954/b048954.txt">Table of n, a(n) for n=1..50</a>
%H David W. Boyd, <a href="http://dx.doi.org/10.1016/0022-247X(82)90250-5">The asymptotic behaviour of the binomial circulant determinant</a>, Journal of Mathematical Analysis and Applications, Volume 86, Issue 1, March 1982, Pages 30-38.
%H E. Brown and M. Chamberland, <a href="https://www.math.vt.edu/people/brown/doc/gen_gauss_gem.pdf">Generalizing Gauss's Gem</a>, Amer. Math. Monthly, 119 (No. 7, 2012), 597-601. - _N. J. A. Sloane_, Sep 07 2012
%H D. Burde and W. A. Moens, <a href="https://arxiv.org/abs/2009.05434">The structure of Lie algebras with a derivation satisfying a polynomial identity</a>, arXiv:2009.05434 [math.RA], 2020.
%H L. Carlitz, <a href="http://dx.doi.org/10.1090/S0002-9939-1959-0108466-8">A determinant connected with Fermat's last theorem</a>, Proc. Amer. Math. Soc. 10 (1959), 686-690.
%H L. Carlitz, <a href="http://dx.doi.org/10.1090/S0002-9939-1960-0117192-9">A determinant connected with Fermat's last theorem</a>, Proc. Amer. Math. Soc. 11 (1960), 730-733.
%H Joshua Cooper and Zhibin Du, <a href="https://arxiv.org/abs/2403.02287">Note on the spectra of Steiner distance hypermatrices</a>, arXiv:2403.02287 [math.CO], 2024. See pp. 2, 4.
%H Greg Fee and Andrew Granville, <a href="http://dx.doi.org/10.1090/S0025-5718-1991-1094948-8">The prime factors of Wendt's binomial circulant determinant</a>, Math. Comp. 57 (1991), 839-848.
%H David Ford and Vijay Jha, <a href="http://projecteuclid.org/euclid.em/1048516216">On Wendt's Determinant and Sophie Germain's Theorem</a>, Experimental Mathematics, 2 (1993) No. 2, 113-120.
%H J. S. Frame, <a href="http://www.fq.math.ca/Scanned/18-1/frame.pdf">Factors of the binomial circulant determinant</a>, Fibonacci Quart., 18 (1980), pp. 9-23.
%H Charles Helou, <a href="http://dx.doi.org/10.1090/S0025-5718-97-00870-3">On Wendt's Determinant</a>, Math. Comp., 66 (1997) No. 219, 1341-1346.
%H Charles Helou and Guy Terjanian, <a href="http://dx.doi.org/10.1016/j.jnt.2004.09.003">Arithmetical properties of wendt's determinant</a>, Journal of Number Theory, Volume 115, Issue 1, November 2005, Pages 45-57.
%H Emma Lehmer, <a href="http://dx.doi.org/10.1090/S0002-9904-1935-06210-X">On a resultant connected with Fermat's last theorem</a>, Bull. Amer. Math. Soc. 41 (1935), 864-867.
%H Gerard P. Michon, <a href="http://www.numericana.com/data/wendt.htm">Factorization of Wendt's Determinant</a> (table for n=1 to 114)
%H Anastasios Simalarides, <a href="http://dx.doi.org/10.1090/S0025-5718-00-01292-8">Upper bounds for the prime divisors of Wendt's determinant</a>, Math. Comp., 71 (2002), 415-427.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CirculantMatrix.html">Circulant matrix</a>
%H E. Wendt, <a href="http://eudml.org/doc/204299">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</a>, Reimer (Berlin), 1894.
%F a(2*n) = A129205(n)^2 * (1-4^n).
%F 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
%F a(n) = (-1)^(n-1) * (2^n - 1) * A215615(n)^2. - _Jonathan Sondow_, Aug 17 2012
%F a(2*n) = -3 * A215616(n)^3. - _Jonathan Sondow_, Aug 18 2012
%e a(2) = det [ 1 2 ; 2 1 ] = -3.
%e a(3) = det [ 1 3 3 ; 3 1 3 ; 3 3 1 ] = 28.
%e a(4) = det [ 1 4 6 4 ; 4 1 4 6 ; 6 4 1 4 ; 4 6 4 1 ] = -375.
%t a[ n_] := Resultant[ x^n - 1, (1+x)^n - 1, x];
%o (PARI) {a(n) = if( n<1, 0, matdet( matrix( n, n, i, j, binomial( n, (j-i)%n ))))}
%o (PARI) a(n) = polresultant( x^n - 1, (1+x)^n - 1, x )
%Y Cf. A052182 (circulant of natural numbers), A066933 (circulant of prime numbers), A086459 (circulant of powers of 2), A086569, A129205, A215615, A215616.
%Y See A096964 for another definition.
%K sign,nice
%O 1,2
%A _Eric W. Weisstein_
%E Additional comments from _Michael Somos_, May 27 2000 and Dec 16 2001
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