An $a$ is a sequence of integers of length $k$ such that each term $a_{i}$ for $0 (with $a_{0}=1$) is the sum of two previous terms in at least one way. In the sum $a_{m}+a_{n}$ it is not required that $m\neq n$. For example, 1, 2, 3, 5, 10, 20, 40, 80, is an addition chain of length 7: 3 is 1 + 2, 5 = 2 + 3, 10 = 5 + 5, and the rest have $m=n$.