  We study the cyclic presentations with relators of the form
$x_ix_{i+m}x_{i+k}^{-1}$ and the groups they define. These "groups of Fibonacci
type" were introduced by Johnson and Mawdesley and they generalize the
Fibonacci groups $F(2,n)$ and the Sieradski groups $S(2,n)$. With the exception
of two groups, we classify when these groups are fundamental groups of
3-manifolds, and it turns out that only Fibonacci, Sieradski, and cyclic groups
arise. Using this classification, we completely classify the presentations that
are spines of 3-manifolds, answering a question of Cavicchioli, Hegenbarth, and
Repov\v{s}. When $n$ is even the groups $F(2,n),S(2,n)$ admit alternative
cyclic presentations on $n/2$ generators. We show that these alternative
presentations also arise as spines of 3-manifolds.
