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A048954
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Wendt determinant of n-th circulant matrix C(n).
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12
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1, -3, 28, -375, 3751, 0, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, 0, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 (noe(AT)sspectra.com), Jul 21 2003
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REFERENCES
| P. Ribenboim, "Fermat's Last Theorem for Amateurs", Springer-Verlag, NY, 1999, pp. 126, 136.
Anastasios Simalarides, "Upper bounds for the prime divisors of Wendt's determinant", Math. Comp., 71(2002),415-427.
P. Ribenboim, 13 Lectures on Fermat's last theorem, Springer-Verlag, NY, 1979, pp. 61-63. MR0551363 (81f:10023)
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..50
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114) [From Gerard P. Michon (g.michon(AT)att.net), Jan 16 2009]
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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
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MATHEMATICA
| a[n_] := Resultant[x^n-1, (1+x)^n-1, x]
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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 ))))}
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CROSSREFS
| Cf. A052182 (circulant of natural numbers), A066933 (circulant of prime numbers), A086459 (circulant of powers of 2), A086569.
See A096964 for another definition.
A129205(n)^2*(1-4^n) = a(2*n).
Sequence in context: A072343 A151423 A161605 * A086569 A143636 A060545
Adjacent sequences: A048951 A048952 A048953 * A048955 A048956 A048957
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KEYWORD
| sign,nice
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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EXTENSIONS
| Additional comments from Michael Somos, May 27 2000 and Dec 16 2001
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