 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 isstarting with $N_{i,1}(x)=1_{[t_i,t_{i+1})}(x)$. Terms with zero denominator are interpreted as zero in the usual repeated-knot convention. For the uniform knots here, $t_i=i$, this reduces toThe index of the second lower-order B-spline is $i+1$, not $i-1$. Both lower-order supports combine to give $[i,i+k]$.