A384313 a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(0), F(1), ..., F(n-1)), where F = A000045 (Fibonacci numbers), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

0, 0, 2, 9, 582, 27136, 7661772, 2797055478, 4374706319136, 11681281664592429, 112352959301265272414, 2147474541377915674682880, 133430162305143400794479937840, 18069411470335957872130103264497774, 7436752857750595469877425837627133763584

Offset: 1

Author

Clark Kimberling, Jun 27 2025

Keywords

• nonn

Examples

			The rows of M(4) are (0,1,1,2), (2,0,1,1), (1,2,0,1), (1,1,2,0); determinant(M(4)) = -16; permanent(M(4)) = 34, so neg(M(4)) = (-16 - 34)/2 = -25 and pos(M(4)) = (-16 + 34)/2 = 9.
		

Crossrefs

Cf. A123744 (determinant), A384079 (permanent), A380661, A384080.

Programs

Mathematica

z = 14;
v[n_] := Table[Fibonacci[k], {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
p = Table[Permanent[u[n]], {n, 1, z}]   (* A384079 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123744 *)
neg = (d - p)/2    (* A384080 *)
pos = (d + p)/2    (* A384313 *)

Formula

a(n) = (-(-1)^n * A123744(n) + A384079(n)) / 2.