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 A123744 Circulants of Fibonacci numbers (including F_0 = 0). 3

%I

%S 0,1,2,16,287,16128,2192140,830952837,805644641664,2080690769701456,

%T 14002804169885909807,247753675148653634781184,

%U 11469641168045182197979378136,1391545878431673359565624090480585,442017027765434652128920030338417270784,367683484076057642925500106042968712221296320

%N Circulants of Fibonacci numbers (including F_0 = 0).

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

%F 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 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 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 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 n=4: a(4)= Det(M(4)) = 16 = (1-I)*2*(1+I)*4.

%o (PARI) mm(n) = matdet(matrix(n, n, i, j, fibonacci(n-1-lift(Mod(j-i, n))))); \\ _Michel Marcus_, Aug 11 2019

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

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

%K nonn,easy

%O 1,3

%A _Wolfdieter Lang_, Nov 10 2006, Jan 27 2009

%E More terms from _Michel Marcus_, Aug 11 2019

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Last modified July 13 10:38 EDT 2020. Contains 335687 sequences. (Running on oeis4.)