= Solution
Collecting the shifted terms gives a <second-order difference equation> with coefficients $1-h/2$, $-(2+6h^2)$ and $1+h/2$. Substitute $y_n=r^n$ to obtain its <characteristic equation of a linear recurrence>:
$$
(1-h/2)r^2-(2+6h^2)r+(1+h/2)=0.
$$
For real $h\ne0,2$, the roots and <general solution> are
$$
r_\pm=\frac{1+3h^2\pm\frac h2\sqrt{25+36h^2}}{1-h/2},\qquad \boxed{y_n=Ar_+^n+Br_-^n.}
$$
In the small-step regime, indeed for $0<h<2$, the <polynomial> is positive at $r=0$, negative at $r=1$ and positive for sufficiently large positive $r$. Thus $0<r_-<1<r_+$. The <bounded solution of a second-order constant-coefficient recurrence> discards the growing root, and $y_0=1$ fixes its remaining coefficient:
$$
\boxed{y_n=r_-^n,\qquad r_-=1-2h+2h^2+O(h^3),\qquad y_n\approx(1-2h)^n.}
$$
The <Taylor series> establishes the displayed approximation to the root. Raising that approximation to the $n$th power is a leading approximation, rather than a uniform relative approximation for arbitrarily large $n$: its relative error is controlled when $nh^2$ is small. For completeness, the degenerate values give $y_n=A+Bn$ when $h=0$, and $y_n=C13^{-n}$ when $h=2$.
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