  We consider various properties and manifestations of some sign-alternating
univariate polynomials borne of right-triangular integer arrays related to
certain generalizations of the Fibonacci sequence. Using a theory of the root
geometry of polynomial sequences developed by J. L. Gross, T. Mansour, T. W.
Tucker, and D. G. L. Wang, we show that the roots of these `sign-alternating
Gibonacci polynomials' are real and distinct, and we obtain explicit bounds on
these roots. We also derive Binet-type closed expressions for the polynomials.
Some of these results are applied to resolve finiteness questions pertaining to
a one-player combinatorial game (or puzzle) modelled after a well-known puzzle
we call the `Networked-numbers Game.' Elsewhere, the first- and second-named
authors, in collaboration with A. Nance, have found rank symmetric
`diamond-colored' distributive lattices naturally related to certain
representations of the special linear Lie algebras. Those lattice cardinalities
can be computed using sign-alternating Fibonacci polynomials, and the lattice
rank generating functions correspond to the rows of some new and easily defined
triangular integer arrays. Here, we present Gibonaccian, and in particular
Lucasian, versions of those symmetric Fibonaccian lattices/results, but without
the algebraic context of the latter.
