A118704 a(n) = determinant of n X n circulant matrix whose first row is the first n distinct Fibonacci numbers A000045(2), A000045(3), ... A000045(n+1).

1, -3, 18, -429, 24149, -3813376, 1513739413, -1575456727131, 4215561680804992, -29321025953223722025, 529210578655758192641625, -24875949855198086445567836160, 3047957640551011125902187378426905, -974921913036976554924444728974464589255

Offset: 1

Author

Jonathan Vos Post, May 20 2006

Keywords

• easy
• sign

Comments

a(n) alternates in sign.

Examples

			a(2) = -3 because of the determinant -3 =
| 1, 2 |
| 2, 1 |.
a(5) = 24149 = determinant
| 1, 2, 3, 5, 8 |
| 8, 1, 2, 3, 5 |
| 5, 8, 1, 2, 3 |
| 3, 5, 8, 1, 2 |
| 2, 3, 5, 8, 1 |.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Circulant Matrix.

Crossrefs

See also: A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A066933 Circulant of prime numbers. A086459 Circulant of powers of 2.

Cf. A000045, A048954, A052182, A066933, A086459, A086569.

Programs

Maple

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->
        (<<0|1>, <1|1>>^(2+irem(n-i+j, n)))[1, 2])):
seq(a(n), n=1..15);  # Alois P. Heinz, Oct 23 2009

Formula

a(n) ~ (-1)^(n+1) * phi^(n*(n+1)) / 5^(n/2), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 10 2025

Extensions

Corrected and extended by Alois P. Heinz, Oct 23 2009