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Search: id:A086459
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| A086459 |
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Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1,2,4,8,..2^(n-1)) right. |
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+0
10
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| 1, -3, 49, -3375, 923521, -992436543, 4195872914689, -70110209207109375, 4649081944211090042881, -1227102111503512992112190463, 1291749870339606615892191271170049, -5429914198235566686555216227881787109375
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Note that if the rows are rotated left instead of right, the sign of the terms for which n = 0 or 3 (mod 4) is reversed. The n eigenvalues of these circulant matrices lie on the circle of radius 2(2^n - 1)/3 centered at x=(2^n - 1)/3, y=0. This sequence can be generalized to bases other than 2 and similar results are true.
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REFERENCES
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Richard Bellman, Introduction to Matrix Analysis, Second Edition, SIAM, 1970, pp. 242-3.
Philip J. Davis, Circulant Matrices, Second Edition, Chelsea, 1994.
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LINKS
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Eric Weisstein's World of Mathematics, Circulant Matrix
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FORMULA
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a(n) = (-2^n+1)^(n-1)
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EXAMPLE
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a(3) = determinant of the matrix ((1,2,4),(4,1,2),(2,4,1)) = 49. [Corrected by T. D. Noe (noe(AT)sspectra.com), Jan 22 2008]
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MAPLE
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restart:with (combinat):a:=n->mul(-stirling2(n, 2), j=3..n): seq(a(n), n=2..19); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 01 2009]
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MATHEMATICA
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Table[x=2^Range[0, n-1]; m=Table[RotateRight[x, i-1], {i, n}]; Det[m], {n, 12}]
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CROSSREFS
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Cf. A048954 (circulant of binomial coefficients), A052182 (circulant of natural numbers), A066933 (circulant of prime numbers).
Sequence in context: A012223 A012100 A106842 this_sequence A063893 A145572 A034201
Adjacent sequences: A086456 A086457 A086458 this_sequence A086460 A086461 A086462
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KEYWORD
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easy,sign
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Jul 21 2003
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