We analyse the roots of the polynomial $x^n-p{x}^{n-1}-qx-1$ for $p⩾q⩾1$. This is the characteristic polynomial of the recurrence relation $F_{k,p,q}\left(n\right)=p{F}_{k,p,q}(n-1)+q{F}_{k,p,q}(n-k+1)+F_{k,p,q}(n-k)$ for $n⩾k$, which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.

We construct parametric families of (monic) reducible polynomials having two roots very close to each other.

We give a very brief, but gentle, sketch of an introduction both to the Rosen continued fractions and to a geometric setting to which they are related, given in terms of Veech groups. We have kept the informal approach of the talk at the Numerations conference, aimed at an audience assumed to have heard of neither of the topics of the title.The Rosen continued fractions are a family of continued fraction algorithms, each gives expansions of real numbers in terms of elements of a corresponding algebraic...