My teacher wrote the this equation for the Romberg method
$ I_{j,k}=\frac{4^j I_{j-1/k+1}-I_{j-1/k}}{4^j-1} $
Is this the right equation? Most the equations I looked at online for the Romberg equation look different; for example, see Romberg Method and Romberg Method
In general, the Romberg's method can be described as $$ T_m^k=\frac{4^m T_{m-1}^k-T_{m-1}^{k-1}}{4^m-1} $$ where $T_m^k$ means the quadrature result with $k$ times dichotomies and $m$ times extrapolations. Let the integral interval is $[a,b]$, for example, $T_0^0$ means the result of the quadrature with step $b-a$. $T^1_0$ means the result with step $\frac{b-a}{2}$. And $T_1^0$ means the extrapolation result that linear combination of $T_0^0$ and $T_0^1$ (See Richardson extrapolation). So, in your shown formula, the subscripts of $k$ seems wrong because linear combination of $k+1$ and $k$ dichotomy results gives a $k+1$ dichotomy result not a $k$ one.