  We study higher-dimensional interlacing Fibonacci sequences, generated via
both Chebyshev type functions and $m$-dimensional recurrence relations. For
each integer $m$, there exist both rational and integer versions of these
sequences, where the underlying prime congruence structures of the rational
sequence denominators enables the integer sequence to be recovered.
  From either the rational or the integer sequences we construct sequences of
vectors in $\mathbb{Q}^m$, which converge to irrational algebraic points in
$\mathbb{R}^m$. The rational sequence terms can be expressed as simple
recurrences, trigonometric sums, binomial polynomials, sums of squares, and as
sums over ratios of powers of the signed diagonals of the regular unit $n$-gon.
These sequences also exhibit a "rainbow type" quality, and correspond to the
Fleck numbers at negative indices, leading to some combinatorial identities
involving binomial coefficients.
  It is shown that the families of orthogonal generating polynomials defining
the recurrence relations employed, are divisible by the minimal polynomials of
certain algebraic numbers, and the three-term recurrences and differential
equations for these polynomials are derived. Further results relating to the
Christoffel-Darboux formula, Rodrigues' formula and raising and lowering
operators are also discussed. Moreover, it is shown that the Mellin transforms
of these polynomials satisfy a functional equation of the form $p_n(s)=\pm
p_n(1-s)$, and have zeros only on the critical line Re $s=1/2$.
