= Solution
For unit-spaced knots, the <quadratic cardinal B-spline> is a translate of
$$
N_{j,3}(t)=
\begin{cases}
(t-j)^2/2,&j\le t\le j+1,\\
3/4-(t-j-3/2)^2,&j+1\le t\le j+2,\\
(j+3-t)^2/2,&j+2\le t\le j+3,\\
0,&\text{otherwise}.
\end{cases}
$$
For example, the <Cox-de Boor recurrence> gives $N_{j,3}(t)=(t-j)N_{j,2}(t)/2+(j+3-t)N_{j+1,2}(t)/2$ from the two linear hats. Substituting their linear pieces gives the three expressions above. At $x_i=i+3/2$, only the <B-splines> with indices $j=i-1,i,i+1$ have nonzero values, and their values are respectively
$$
\boxed{N_j(x_i)=
\begin{cases}
3/4,&j=i,\\
1/8,&|j-i|=1,\\
0,&|j-i|\ge2.
\end{cases}}
$$
Only indices $1\le j\le n$ belong to the given <basis>, so entries outside that range are omitted at the boundary.
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