A135288 Simple example of tridiagonal one sequence system using the Fibonacci sequence to give a triangular sequence based on the coefficients of the characteristic polynomials.

1, 1, -1, -3, -1, 1, -7, 3, 3, -1, 17, 12, -9, -4, 1, 43, -8, -41, 6, 8, -1, -109, -96, 91, 72, -20, -12

Offset

1,4

References

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

Links

Table of n, a(n) for n=1..27.

Formula

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))

Example

{{1},
{1, -1},
{-3, -1, 1},
{-7, 3, 3, -1},
{17, 12, -9, -4, 1},
{43, -8, -41,6, 8, -1},
{-109, -96, 91, 72, -20, -12, 1}}

Mathematica

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]

Crossrefs

Cf. A124032.

Sequence in context: A228524 A116407 A309402 * A078026 A350635 A126713

Adjacent sequences: A135285 A135286 A135287 * A135289 A135290 A135291

Keyword

uned,sign

Author

Roger L. Bagula, Feb 16 2008

Status

approved