A384593 a(n) = pos(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.

1, 1, 10, 115, 7182, 630400, 236525784, 146492200737, 343945864714624, 1504008427530969535, 22752527891269251753118, 704885020930148344661403136, 70113811460119152088048405685808, 15355281759224121782879344275585373609

Offset

1,3

Links

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

Formula

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

Example

The rows of M(4) are (1,1,2,3), (3,1,1,2), (2,3,1,1), (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.

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 *)

Crossrefs

Cf. A000045, A380661, A123745 (determinant), A384591 (permanent), A384592.

Sequence in context: A233908 A089833 A308667 * A251318 A083448 A024129

Adjacent sequences: A384590 A384591 A384592 * A384594 A384595 A384596

Keyword

nonn

Author

Clark Kimberling, Jul 02 2025

Status

approved