A384596 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(2), F(3), ..., F(n+1)), where F = A000045 (Fibonacci numbers), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

1, 1, 36, 630, 75264, 10309104, 6689940744, 6609305472651, 25797682556382400, 181805125100075828614, 4497447436889592767655228, 225317753810449180044272832000, 36410024238337826166260377355303568, 12904889422278677354475665889049659231531

Offset: 1

Author

Clark Kimberling, Jul 10 2025

Keywords

• nonn

Examples

			The rows of M(4) are (1,2,3,5), (5,1,2,3), (3,5,1,2), (2,3,5,1); determinant(M(4)) = -429; permanent(M(4)) = 1689, so neg(M(4)) = (-429 - 1689)/2 = -1059 and pos(M(4)) = (-429 + 1689)/2 = 630.
		

Crossrefs

Cf. A000045, A118704 (determinant), A384594 (permanent), A384595.

Programs

Mathematica

z = 14;
v[n_] := Table[Fibonacci[k], {k, 2, n + 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
p = Table[Permanent[u[n]], {n, 1, z}]   (* A384594 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A118704 *)
neg = (d - p)/2     (* A384595 *)
pos = (d + p)/2     (* A384596 *)

Formula

a(n) = (1/2)*(A118704(n) + A384594(n)).