A123745 Circulants of Fibonacci numbers (without F_0 = 0).

1, 0, 4, 35, 1812, 170240, 46301673, 30413016864, 52171354014208, 228072747428273319, 2583414317082067853704, 75732718487930382583857152, 5773860969402842827019263155009, 1146353725688692827225795357533033072, 593830518002528577221255815133242142736384

Offset: 1

Author

Wolfdieter Lang, Nov 10 2006, Jan 27 2009

Keywords

• nonn
• easy

Comments

A circulant C_n is the determinant of a circulant n X n matrix M, i.e. one with entries M_{i,j}=a_{i-j} where the indices are taken mod n. Notation: C_n = C_n([a_n,a_{n-1},...,a_1]), with the first row of M given.

The name circulant is (unfortunately) used for matrices as well as for their determinants. The matrix could be called circular instead.

The eigenvalues of a circulant n X n matrix M(n) are lambda^{(n)}_k=sum(a_j*(rho_n)^(j*k),j=1..n), with the n-th roots of unity (rho_n)^k, k=1..n, where rho_n:=exp(2*Pi/n). See the P. J. Davis reference which uses a different convention.

Examples

			n=4: the circulant 4 X 4 matrix is M(4) = matrix([3,2,1,1],[1,3,2,1],[1,1,3,2],[2,1,1,3]).
n=4: 4th roots of unity: rho_4 = I, (rho_4)^2 = -1, (rho_4)^3 = -I, (rho_4)^4 =1, with I^2=-1.
n=4: the eigenvalues of M(4) are therefore 1*I^k + 1*(-1)^k + 2*(-I)^k + 3*1^k, k=1,...,4, namely 2-I,1,2+I,7.
n=4: a(4)= Det(M(4)) = 35 = (2-I)*1*(2+I)*7.
		

References

• P. J. Davis, Circulant Matrices, J. Wiley, New York, 1979.

Crossrefs

Cf. A123744 (Fibonacci circulants including F_0 = 0).

Cf. A052182 (with n instead of Fibonacci(n) and first row reversed).

Programs

PARI

a(n) = matdet(matrix(n, n, i, j, fibonacci(n-lift(Mod(j-i, n))))); \\ Michel Marcus, Aug 11 2019

Formula

a(n) = product(lambda^{(n)}k,k=1..n), with lambda^{(n)}_k=sum(F{j}*(rho_n)^(j*k),j=1..n).

a(n) = C_n([F_{n},F_{n-2},...,F_1]) with the Fibonacci numbers F_n:=A000045(n) and the circulant C_n (see comment above).