A384595 a(n) = neg(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 neg(M(n)) is the negative part of the determinant of M(n); see A380661.

0, -4, -18, -1059, -51115, -14122480, -5176201331, -8184762199782, -21582120875577408, -211126151053299550639, -3968236858233834575013603, -250193703665647266489840668160, -33362066597786815040358189976876663, -13879811335315653909400110618024123820786

Offset

1,2

Links

Table of n, a(n) for n=1..14.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A000045, A118704 (determinant), A384594 (permanent), A384596.

Sequence in context: A342467 A367533 A156491 * A287677 A069874 A222323

Adjacent sequences: A384592 A384593 A384594 * A384596 A384597 A384598

Keyword

sign

Author

Clark Kimberling, Jul 10 2025

Status

approved