cp's OEIS Frontend

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).

%I A118704 #18 Jul 10 2025 16:08:07
%S A118704 1,-3,18,-429,24149,-3813376,1513739413,-1575456727131,
%T A118704 4215561680804992,-29321025953223722025,529210578655758192641625,
%U A118704 -24875949855198086445567836160,3047957640551011125902187378426905,-974921913036976554924444728974464589255
%N 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).
%C A118704 a(n) alternates in sign.
%H A118704 Alois P. Heinz, <a href="/A118704/b118704.txt">Table of n, a(n) for n = 1..50</a>
%H A118704 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantMatrix.html">Circulant Matrix</a>.
%F A118704 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
%e A118704 a(2) = -3 because of the determinant -3 =
%e A118704 | 1, 2 |
%e A118704 | 2, 1 |.
%e A118704 a(5) = 24149 = determinant
%e A118704 | 1, 2, 3, 5, 8 |
%e A118704 | 8, 1, 2, 3, 5 |
%e A118704 | 5, 8, 1, 2, 3 |
%e A118704 | 3, 5, 8, 1, 2 |
%e A118704 | 2, 3, 5, 8, 1 |.
%p A118704 a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->
%p A118704         (<<0|1>, <1|1>>^(2+irem(n-i+j, n)))[1, 2])):
%p A118704 seq(a(n), n=1..15);  # _Alois P. Heinz_, Oct 23 2009
%Y A118704 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.
%Y A118704 Cf. A000045, A048954, A052182, A066933, A086459, A086569.
%K A118704 easy,sign
%O A118704 1,2
%A A118704 _Jonathan Vos Post_, May 20 2006
%E A118704 Corrected and extended by _Alois P. Heinz_, Oct 23 2009