= Solution
Use the standard partition normalization of <B-splines>, with order $k$ meaning degree $k-1$. The <Cox-de Boor recurrence> on arbitrary increasing <spline knots> is
$$
N_{i,k}(x)=\frac{x-t_i}{t_{i+k-1}-t_i}N_{i,k-1}(x)+\frac{t_{i+k}-x}{t_{i+k}-t_{i+1}}N_{i+1,k-1}(x),
$$
starting with $N_{i,1}(x)=\mathbf1_{[t_i,t_{i+1})}(x)$. Terms with zero denominator are interpreted as zero in the usual repeated-knot convention. For the uniform knots here, $t_i=i$, this reduces to
$$
\boxed{N_{i,k}(x)=\frac{x-i}{k-1}N_{i,k-1}(x)+\frac{i+k-x}{k-1}N_{i+1,k-1}(x),\qquad k\ge2.}
$$
The index of the second lower-order <B-spline> is $i+1$, not $i-1$. Both lower-order supports combine to give $[i,i+k]$.
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