  The equilateral triangle is one of the few planar domains where the Dirichlet
and Neumann eigenvalue problems were explicitly determined, by Lam\'e in 1833,
despite not admitting separation of variables. In this paper, we study the
Robin spectrum of the equilateral triangle, which was determined by McCartin in
2004 in terms of a system of transcendental coupled secular equations.
  We give uniform upper bounds for the Robin-Neumann gaps, showing that they
are bounded by their limiting mean value, which is hence an almost sure bound.
The spectrum admits a systematic double multiplicity, and after removing it we
study the gaps in the resulting desymmetrized spectrum. We show a spectral gap
property, that there are arbitrarily large gaps, and also arbitrarily small
ones, moreover that the nearest neighbour spacing distribution of the
desymmetrized spectrum is a delta function at the origin. We show that for
sufficiently small Robin parameter, the desymmetrized spectrum is simple.
