  We present a family of rank symmetric diamond-colored distributive lattices
that are naturally related to the Fibonacci sequence and certain of its
generalizations. These lattices re-interpret and unify descriptions of some un-
or differently-colored lattices found variously in the literature. We
demonstrate that our symmetric Fibonaccian lattices naturally realize certain
(often reducible) representations of the special linear Lie algebras, with
weight basis vectors realized as lattice elements and Lie algebra generators
acting along the order diagram edges of each lattice. We present evidence that
each such weight basis is, in a certain sense, uniquely associated with its
lattice. We provide new descriptions of the lattice cardinalities and rank
generating functions and offer several conjectures/open problems. Throughout,
we make connections with integer sequences from the OEIS.
