A123744 Circulants of Fibonacci numbers (including F_0 = 0).

1, 0, 1, 2, 16, 287, 16128, 2192140, 830952837, 805644641664, 2080690769701456, 14002804169885909807, 247753675148653634781184, 11469641168045182197979378136, 1391545878431673359565624090480585, 442017027765434652128920030338417270784, 367683484076057642925500106042968712221296320

Offset: 0

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. Hence C_n=C_n([a_n,a_{n-1},...,a_1]), with the first row of M given.

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 circular 4 X 4 matrix is M(4) = matrix([[2,1,1,0],[0,2,1,1],[1,0,2,1],[1,1,0,2]]).
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. A123744 n=4: the eigenvalues of M(4) are therefore: 0*I^k + 1*(-1)^k + 1*(-I)^k + 2*1^k, k=1,...,4, namely 1-I, 2, 1+I, 4.
n=4: a(4)= Det(M(4)) = 16 = (1-I)*2*(1+I)*4.
		

References

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

Crossrefs

Cf. A123745 (other Fibonacci circulants without F_0 = 0).

Cf. A081131 (with n instead of Fibonacci(n)).

Cf. A000045.

Programs

PARI

mm(n) = matdet(matrix(n, n, i, j, fibonacci(n-1-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-1}*(rho_n)^(j*k),j=1..n).

a(n) = C_n([F_{n-1},F_{n-2},...,F_0]) with the Fibonacci numbers F_n:=A000045(n) and the circulant C_n (see comment above).

Extensions

More terms from Michel Marcus, Aug 11 2019

a(0)=1 prepended by Alois P. Heinz, Jun 27 2025