What is the connection of the sequence 3, 4, 5/3, 2/3, 1 with deep topics?
Solution 1:
Hyperbolic geometry / algebraic K-theory must mean that $(1+x)/y$ and its iterates appear in the functional equations of one of the dilogarithm functions (Abel's relation or similar). There is a 5-term functional equation with some cyclic symmetry.
Models of QFT may refer to the "pentagon relation" from integrable models. This again reflects the order 5 property.
[added:]
This does appear to be what Zagier was pointing to. Back in the 1990's,
ADE functional dilogarithm identities and integrable models F. Gliozzi, R. Tateo http://xxx.lanl.gov/abs/hep-th/9411203
formulated a functional equation generalizing some identities for the Rogers dilogarithm evaluated at roots of unity. The latter identities were known to correspond to torsion in $K$-theory of fields (see papers of Frenkel and Szenes on dilogarithm identities, also early and mid-90's, on arxiv, including proofs of Gliozzi-Tateo formulae and some comments on $K_3(\mathbb{C})$).
On page 4-5 of the paper it is pointed out that the 5-fold periodicity comes from an inspection of the cyclic symmetry of the Abel 5-term functional equation of the Rogers dilogarithm. If the terms are numbered correctly then $(1+x)/y$ is the generator of this symmetry. The periodicity of this transformation was probably observed about 200 years ago.
Cluster algebras appeared later and provided the technology to prove (among many other things) more general periodicity conjectures from integrable models of QFT, going back to earlier papers of Zamolodchikov. However, the connection of this one period-5 transformation to the dilogarithm is probably easiest to see in the Gliozzi/Tateo paper above. K-theory and hyperbolic geometry have no known direct connection to cluster algebras, though the latter are related to triangulations of polygons and this is sometimes connected to hyperbolic 2- and 3-dimensional geometry.
Solution 2:
Google "cluster algebra", e.g. see this paper Combinatorial interpretations for rank-two cluster algebras of affine type.