  Continued fractions are linked to Stern's diatomic sequence
0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation a_2n=a_n and a_{2n+1}
= a_n + a_{n+1}, where a_0=0 and a_1=1), which has long been known. Using a
particular multidimensional continued fraction algorithm (the Farey algorithm),
we will generalize the diatomic sequence to a collection of numbers that quite
naturally should be called the triatomic sequence (or a two-dimensional Pascal
with memory sequence). As continued fractions and the diatomic sequence can be
thought of as coming from systematic subdivisions of the unit interval, this
new triatomic sequence will arise by a systematic subdivision of a triangle. We
will discuss some of the algebraic properties for the tri-atomic sequence.
