Search: id:A048954
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%I A048954
%S A048954 1,3,28,375,3751,0,6835648,1343091375,364668913756,210736858987743,
%T A048954 101832157445630503,0,487627751563388801409591,
%U A048954 4875797582053878382039400448,58623274842128064372315087290368
%V A048954 1,-3,28,-375,3751,0,6835648,-1343091375,364668913756,-210736858987743,
%W A048954 101832157445630503,0,487627751563388801409591,
%X A048954 -4875797582053878382039400448,58623274842128064372315087290368
%N A048954 Wendt determinant of n-th circulant matrix C(n).
%C A048954 det(C(n))=0 for n divisible by 6.
%C A048954 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
%D A048954 P. Ribenboim, "Fermat's Last Theorem for Amateurs", Springer-Verlag, 
               NY, 1999, pp. 126, 136.
%D A048954 Anastasios Simalarides, "Upper bounds for the prime divisors of Wendt's 
               determinant", Math. Comp., 71(2002),415-427.
%D A048954 P. Ribenboim, 13 Lectures on Fermat's last theorem, Springer-Verlag, 
               NY, 1979, pp. 61-63. MR0551363 (81f:10023)
%H A048954 T. D. Noe, <a href="b048954.txt">Table of n, a(n) for n=1..50</a>
%H A048954 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CirculantMatrix.html">Link to a section of The World of Mathematics.</
               a>
%H A048954 Gerard P. Michon, <a href="http://www.numericana.com/data/wendt.htm">
               Factorization of Wendt's Determinant</a> (table for n=1 to 114) [From 
               Gerard P. Michon (g.michon(AT)att.net), Jan 16 2009]
%F A048954 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
%t A048954 a[n_] := Resultant[x^n-1, (1+x)^n-1, x]
%o A048954 (PARI) a(n)=if(n<1,0,matdet(matrix(n,n,i,j,binomial(n,(j-i)%n))))
%o A048954 (PARI) {a(n)= if(n<1, 0, matdet( matrix( n, n, i, j, binomial( n, (j-i)%n 
               ))))}
%Y A048954 Cf. A052182 (circulant of natural numbers), A066933 (circulant of prime 
               numbers), A086459 (circulant of powers of 2), A086569.
%Y A048954 See A096964 for another definition.
%Y A048954 A129205(n)^2*(1-4^n) = a(2*n).
%Y A048954 Sequence in context: A072343 A151423 A161605 this_sequence A086569 A143636 
               A060545
%Y A048954 Adjacent sequences: A048951 A048952 A048953 this_sequence A048955 A048956 
               A048957
%K A048954 sign,nice
%O A048954 1,2
%A A048954 Eric Weisstein (eric(AT)weisstein.com)
%E A048954 Additional comments from Michael Somos, May 27 2000 and Dec 16 2001

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