Companion Coordinate Examples

Companion coordinates form one part of the general coordinate atlas described in General Coordinate Atlas.

A constant-coefficient recurrence of order \(d\) provides a companion state model. Every finite sequence can be represented by its initial state and unique defect stream. This applies to arbitrary custom recurrences and to the named Fibonacci, Lucas, Pell, Jacobsthal, Tribonacci, and Padovan models.

The Euler-to-Fibonacci construction shows why defect coordinates matter. Signed secant Euler numbers do not satisfy the homogeneous Fibonacci recurrence. Their exact Fibonacci companion representation therefore consists of the initial state and the forcing sequence

\[ \delta_n=\mathcal E_{n+2}-\mathcal E_{n+1}-\mathcal E_n. \]

For arbitrary bases, polynomial families, combinatorial transforms, affine and formal charts, finite fields, state-space systems, tensor transport, and spectral or packet coordinates, use the general atlas documentation.