  In this paper we present exact calculations of the partition function $Z$ of
the $q$-state Potts model and its generalization to real $q$, for arbitrary
temperature on $n$-vertex strip graphs, of width $L_y=2$ and arbitrary length,
of the triangular lattice with free, cyclic, and M\"obius longitudinal boundary
conditions. These partition functions are equivalent to Tutte/Whitney
polynomials for these graphs. The free energy is calculated exactly for the
infinite-length limit of the graphs, and the thermodynamics is discussed.
Considering the full generalization to arbitrary complex $q$ and temperature,
we determine the singular locus ${\cal B}$ in the corresponding ${\mathbb C}^2$
space, arising as the accumulation set of partition function zeros as $n \to
\infty$. In particular, we study the connection with the T=0 limit of the Potts
antiferromagnet where ${\cal B}$ reduces to the accumulation set of chromatic
zeros. Comparisons are made with our previous exact calculation of Potts model
partition functions for the corresponding strips of the square lattice. Our
present calculations yield, as special cases, several quantities of
graph-theoretic interest.
