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\begin{abstract}
Many familiar counting sequences, such as the Catalan, Motzkin, 
Schr\"{o}der and Delannoy numbers, have a generating function 
that is algebraic of degree 2. For example, the GF for the 
central Delannoy numbers is $\frac{1}{\sqrt{1-6x+x^{2}}}$. Here 
we determine all generating functions
of the form $\frac{1}{\sqrt{1+Ax+Bx^{2}}}$ 
that yield counting sequences and 
point out that they have a unified combinatorial 
interpretation in terms of colored lattice paths. We do likewise 
for the related forms $1-\sqrt{1+Ax+Bx^{2}}$ and 
$\frac{1+Ax-\sqrt{1+2Ax+Bx^{2}}}{2Cx^{2}}$.
\end{abstract}

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