  Consider the sequence $\mathcal{V}(2,n)$ constructed in a greedy fashion by
setting $a_1 = 2$, $a_2 = n$ and defining $a_{m+1}$ as the smallest integer
larger than $a_m$ that can be written as the sum of two (not necessarily
distinct) earlier terms in exactly one way; the sequence $\mathcal{V}(2,3)$,
for example, is given by $$ \mathcal{V}(2,3) = 2,3,4,5,9,10,11,16,22,\dots$$ We
prove that if $n \geqslant 5$ is odd, then the sequence $\mathcal{V}(2,n)$ has
exactly two even terms $\left\{2,2n\right\}$ if and only if $n-1$ is not a
power of 2. We also show that in this case, $\mathcal{V}(2,n)$ eventually
becomes a union of arithmetic progressions. If $n-1$ is a power of 2, then
there is at least one more even term $2n^2 + 2$ and we conjecture there are no
more even terms. In the proof, we display an interesting connection between
$\mathcal{V}(2,n)$ and Sierpinski Triangle. We prove several other results,
discuss a series of striking phenomena and pose many problems. This relates to
existing results of Finch, Schmerl & Spiegel and a classical family of
sequences defined by Ulam.
