A384592 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), where F = A000045 (Fibonacci numbers), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

0, -1, -6, -150, -5370, -800640, -190224111, -176905217601, -291774510700416, -1732081174959242854, -20169113574187183899414, -780617739418078727245260288, -64339950490716309261029142530799, -16501635484912814610105139633118406681

Offset: 1

Author

Clark Kimberling, Jul 02 2025

Keywords

• sign

Examples

			The rows of M(4) are (1,1,2,3), (3,1,1,2), (23,1,1), (1,2,3,1); determinant(M(4)) = -35; permanent(M(4)) = 265, so neg(M(4)) = (-35 -265)/2 = -150 and pos(M(4)) = (-35 + 265)/2 = 115.
		

Crossrefs

Cf. A000045, A380661, A123745 (determinant), A384591 (permanent), A384593.

Programs

Mathematica

z = 14;
v[n_] := Table[Fibonacci[k], {k, 1, n}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]]
Table[Permanent[u[n]], {n, 1, z}]   (* A384591  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A123745, with alternating signs *)
neg = (d - p)/2     (* A384592 *)
pos = (d + p)/2     (* A384593 *)

Formula

a(n) = (1/2)*((-1)^n A123745(n) - A384591(n)).