Continued fractions are linked to Stern's diatomic sequence 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, ... (given by the recursion relations α_2n = α_n and α_2n+1 = α_n + α_n+1, where α₀ = 0 and α₁ = 1), which has long been known. Using a particular multidimensional continued fraction algorithm (the Farey algorithm), we generalize the diatomic sequence to a sequence of numbers that quite naturally can be termed Stern's triatomic sequence (or a two-dimensional Pascal sequence with memory). As both continued fractions and the diatomic sequence can be thought of as coming from a systematic subdivision of the unit interval, this new triatomic sequence arises by a systematic subdivision of a triangle. We discuss some of the algebraic properties of the triatomic sequence.