A370724 - OEIS

%I #15 Feb 28 2024 13:26:49

%S 1,0,-1,1,-1,2,0,3,3,27,0,4,0,5,5,198,0,618,7,7,0,8,0,9,2295,74349,0,

%T 18970,-28160,11,-178717,12,0,57132,-13,13,-3247328,14,-1580068,

%U 122895,-94575,19711504,0,111678928,-4001885,5302759,-4875304727,18,0,194717339,19

%N Determinant of n X n Fibonacci word matrix.

%C Take the first n terms of A003849 and form an n X n matrix by rotating these n terms 1 place to the left in each new row (i.e., a circulant matrix).  This sequence is the determinant of the resulting matrix.

%F Conjectures:  a(F(n)) = +-F(n-2) where F(n) is the n-th Fibonacci number, and a(L(n)) = +-L(n-2) where L(n) is the n-th Lucas number.

%e For n=5 the first 5 terms are [0,1,0,0,1] and the resulting matrix is [[0,1,0,0,1],[1,0,1,0,0],[0,1,0,1,0],[0,0,1,0,1],[1,0,0,1,0]].  Its determinant is 2.

%Y Cf. A000032, A000045, A003849, A370725.

%K sign

%O 0,6

%A _Jeffrey Shallit_, Feb 28 2024