Two-parameter families of uniquely extendable Diophantine triples
  Let A,K be positive integers and u=-2,-1,1 or 2. The main contribution of the paper is a proof that each of the D(u^2)-triples {K,A^2K+2uA,(A+1)^2K+2u(A+1)} has unique extension to a D(u^2)-quadruple. 
Mihai Cipu, Yasutsugu Fujita and Maurice Mignotte
math.NT