1,3

A circulant C_n is the determinant of a circular 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_0,a_{n-1},...,a_1]), with the first row of M given. The second row is [a_1,a_0,a_{n-1},...,a_2], etc.

The eigenvalues of a circular 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, and the term circulant for circular.

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

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

a(n) = C_n([Ca_{n-1},Ca_{n-2},...,Ca_0]) with the Catalan numbers Ca_n:=A000108(n), and the circulant C_n (see comment above).

n=4: the circular 4 X 4 matrix is M(4) = matrix([[5,2,1,1],[1,5,2,1],[1,1,5,2],[2,1,1,5]]).

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 + 5*1^k, k=1,..,4, namely 4-I, 3, 4+I, 9.

n=4: a(4)= Det(M(4)) = (4-I)*3*(4+I)*9 = 459.

A123744, A123745 (circulants for Fibonacci numbers).

Sequence in context: A034985 A104690 A035483 this_sequence A053292 A053963 A053941

Adjacent sequences: A145567 A145568 A145569 this_sequence A145571 A145572 A145573

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Feb 05 2009