  We consider the strength and effective content of restricted versions of
Hindman's Theorem in which the number of colors is specified and the length of
the sums has a specified finite bound. Let $\mathsf{HT}^{\leq n}_k$ denote the
assertion that for each $k$-coloring $c$ of $\mathbb{N}$ there is an infinite
set $X \subseteq \mathbb{N}$ such that all sums $\sum_{x \in F} x$ for $F
\subseteq X$ and $0 < |F| \leq n$ have the same color. We prove that there is a
computable $2$-coloring $c$ of $\mathbb{N}$ such that there is no infinite
computable set $X$ such that all nonempty sums of at most $2$ elements of $X$
have the same color. It follows that $\mathsf{HT}^{\leq 2}_2$ is not provable
in $\mathsf{RCA}_0$ and in fact we show that it implies $\mathsf{SRT}^2_2$ in
$\mathsf{RCA}_0$. We also show that there is a computable instance of
$\mathsf{HT}^{\leq 3}_3$ with all solutions computing $0'$. The proof of this
result shows that $\mathsf{HT}^{\leq 3}_3$ implies $\mathsf{ACA}_0$ in
$\mathsf{RCA}_0$.
