A384313 -id:A384313 - cp's OEIS frontend

A384080 a(n) = neg(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 neg(M(n)) is the negative part of the determinant of M(n); see A380661.

0, -1, 0, -25, -295, -43264, -5469632, -3628008315, -3569061677472, -13761972434293885, -98350155131379362607, -2395228216526569309464064, -121960521137098218596500559704, -19460957348767631231695727354978359, -6994735829985160817748505807288716492800

Offset: 1

Clark Kimberling, Jun 19 2025

sign

			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.

Cf. A123744 (determinant), A384079 (permanent), A380661, A384313.

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, with alternating signs *)
neg = (d - p)/2    (* A384080 *)
pos = (d + p)/2    (* A384313 *)

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

A384591 a(n) = permanent of the n X n circulant matrix with (row 1) = (F(1), F(2), ..., F(n)), where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 2, 16, 265, 12552, 1431040, 426749895, 323397418338, 635720375415040, 3236089602490212389, 42921641465456435652532, 1485502760348227071906663424, 134453761950835461349077548216607, 31856917244136936392984483908703780290

Offset: 0

Clark Kimberling, Jun 27 2025

nonn

Cf. A000045, A123744 (determinant), A384080, A384313.

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}]

cp's OEIS Frontend

A384080 a(n) = neg(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 neg(M(n)) is the negative part of the determinant of M(n); see A380661.

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