

A086459


Determinant of the circulant matrix whose rows are formed by successively rotating the vector (1, 2, 4, 8, ..., 2^(n1)) right.


11



1, 3, 49, 3375, 923521, 992436543, 4195872914689, 70110209207109375, 4649081944211090042881, 1227102111503512992112190463, 1291749870339606615892191271170049, 5429914198235566686555216227881787109375
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OFFSET

1,2


COMMENTS

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.


REFERENCES

Richard Bellman, Introduction to Matrix Analysis, Second Edition, SIAM, 1970, pp. 2423.
Philip J. Davis, Circulant Matrices, Second Edition, Chelsea, 1994.


LINKS

Table of n, a(n) for n=1..12.
Eric Weisstein's World of Mathematics, Circulant Matrix


FORMULA

a(n) = (2^n + 1)^(n1).
See formulas in A180602, an unsigned version of this sequence with offset 0. [Paul D. Hanna, Sep 11 2010]


EXAMPLE

a(3) = determinant of the matrix ((1,2,4),(4,1,2),(2,4,1)) = 49. [Corrected by T. D. Noe, Jan 22 2008]


MAPLE

restart:with (combinat):a:=n>mul(stirling2(n, 2), j=3..n): seq(a(n), n=2..19); # Zerinvary Lajos, Jan 01 2009


MATHEMATICA

Table[x=2^Range[0, n1]; m=Table[RotateRight[x, i1], {i, n}]; Det[m], {n, 12}]


CROSSREFS

Cf. A048954 (circulant of binomial coefficients), A052182 (circulant of natural numbers), A066933 (circulant of prime numbers).
Cf. A180602 (unsigned, offset 0). [Paul D. Hanna, Sep 11 2010]
Sequence in context: A298697 A326218 A203743 * A180602 A326342 A203700
Adjacent sequences: A086456 A086457 A086458 * A086460 A086461 A086462


KEYWORD

easy,sign


AUTHOR

T. D. Noe, Jul 21 2003


STATUS

approved



