Periodic binary sequences with the "trinomial property" are considered. Som
e necessary and sufficient conditions for "trinomial pairs" of a nonlinear
sequence of period 2(n) - 1 as well as classifications for trinomial pairs
are derived. Complete searches for trinomial pairs of sequences have been c
ompleted for 3 less than or equal to n less than or equal to 17, We list th
em here for n less than or equal to 12.