A135288 - OEIS

%I #5 Mar 31 2024 15:12:17

%S 1,1,-1,-3,-1,1,-7,3,3,-1,17,12,-9,-4,1,43,-8,-41,6,8,-1,-109,-96,91,

%T 72,-20,-12

%N Simple example of tridiagonal one sequence system using the Fibonacci sequence to give a triangular sequence based on the coefficients of the characteristic polynomials.

%D Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334.

%F Main diagonal of matrices: M(n) a(n)=Fibonacci[n] a0(n,m)=if [m=1,a(n), else (-1)^(n+1) Symmetrical subdiagonal: b(n)=1 t(n,m)=Coefficients of polynomials of(M(n))

%e {{1},

%e {1, -1},

%e {-3, -1, 1},

%e {-7, 3, 3, -1},

%e {17, 12, -9, -4, 1},

%e {43, -8, -41,6, 8, -1},

%e {-109, -96, 91, 72, -20, -12, 1}}

%t Clear[A, a] (* A124032*) a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1] + a[n - 2]; A[1] = {{a[1]}}; A[2] = {{a[2], 1}, {1, -1}}; A[3] = {{a[3], 1, 0}, {1, -1, 1}, {0, 1, 1}}; A[4] = {{a[4], 1, 0, 0}, {1, -1, 1, 0}, {0, 1, 1, 1}, {0, 0, 1, -1}}; A[5] = {{a[5], 1, 0, 0, 0}, {1, -1, 1, 0, 0}, {0, 1, 1, 1, 0}, {0, 0, 1, -1, 1}, {0, 0, 0, 1, 1}}; A[6] = {{a[6], 1, 0, 0, 0, 0}, {1, -1, 1, 0, 0, 0}, {0, 1, 1, 1, 0, 0}, {0, 0, 1, -1, 1, 0}, {0, 0, 0, 1, 1, 1}, {0, 0, 0, 0, 1, -1}}; TableForm[Table[Inverse[A[n]], {n, 1, 6}]]; Join[{a[0]}, Table[CharacteristicPolynomial[A[n], x], {n, 1, 6}]]; a0 = Join[{a[0]}, Table[CoefficientList[CharacteristicPolynomial[A[n], x], x], {n, 1, 6}]]; Flatten[a0]

%Y Cf. A124032.

%K uned,sign

%O 1,4

%A _Roger L. Bagula_, Feb 16 2008