%I A123744
%S A123744 0,1,2,16,287,16128,2192140,830952837,805644641664,2080690769701456,
%T A123744 14002804169885909807,247753675148653634781184,
%U A123744 11469641168045182197979378136,1391545878431673359565624090480585
%N A123744 Circulants of Fibonacci numbers (including F_0 = 0).
%C A123744 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.
%C A123744 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.
%D A123744 P. J. Davis, Circulant Matrices, J. Wiley, New York, 1979.
%F A123744 a(n)=product(lambda^{(n)}_k,k=1..n), with lambda^{(n)}_k=sum(F_{j-1}*(rho_n)^(j*k),
               j=1..n).
%F A123744 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).
%e A123744 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]]).
%e A123744 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.
%e A123744 n=4: a(4)= Det(M(4)) = 16 = (1-I)*2*(1+I)*4.
%Y A123744 Cf. A123745 (other Fibonacci circulants without F_0 = 0).
%Y A123744 Sequence in context: A009764 A000182 A102599 this_sequence A136796 A055546 
               A009549
%Y A123744 Adjacent sequences: A123741 A123742 A123743 this_sequence A123745 A123746 
               A123747
%K A123744 nonn,easy
%O A123744 1,3
%A A123744 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Nov 10 2006, 
               Jan 27 2009