  We implement a new and accurate numerical entropic scheme to investigate the
first-order transition features of the triangular Ising model with
nearest-neighbor ($J_{nn}$) and next-nearest-neighbor ($J_{nnn}$)
antiferromagnetic interactions in ratio $R=J_{nn}/J_{nnn}=1$. Important aspects
of the existing theories of first-order transitions are briefly reviewed,
tested on this model, and compared with previous work on the Potts model. Using
lattices with linear sizes $L=30,40,...,100,120,140,160,200,240,360$ and 480 we
estimate the thermal characteristics of the present weak first-order
transition. Our results improve the original estimates of Rastelli et al. and
verify all the generally accepted predictions of the finite-size scaling theory
of first-order transitions, including transition point shifts, thermal, and
magnetic anomalies. However, two of our findings are not compatible with
current phenomenological expectations. The behavior of transition points,
derived from the number-of-phases parameter, is not in accordance with the
theoretically conjectured exponentially small shift behavior and the well-known
double Gaussian approximation does not correctly describe higher correction
terms of the energy cumulants. It is argued that this discrepancy has its
origin in the commonly neglected contributions from domain wall corrections.
