Direct form 1 filter equation for a biquad filter
This may be a misunderstanding based on a fixed point implementation. Fixed point implementations can benefit from &quot;fractional multiplies&quot; which requires a number format where all absolute values are  1 or less. In Q notation that would be Q15.0 for a signed 16-bit integer. Many processors have a dedicated hardware instruction for this operation, which makes it fast and efficient.
Assuming that all poles and zeros are inside the unit circle, we have $|b_0|, |b_2|, |a_2| &lt; 1$ but $|b_1|, |a_1| &lt; 2$. In order to avoid shifting associated with mixed Q notation, the $b_1$ and $a_1$ coefficients are split in have, so numbers can be kept in the same Q. Hence it would be implemented as
$$y[n] = b_0 \cdot x[n] + b_1/2 \cdot x[n-1] + b_1/2 \cdot x[n-1] + b_2 \cdot x[n-1] ...$$
or
$$y[n] = b_0 \cdot x[n] + (b_1/2 \cdot x[n-1]) \ll 1 + b_2 \cdot x[n-1] ...$$
where $\ll$ is the left-shift operator.
This may looks like the coefficient is multiplied by two but in reality it's first split in half to make it's magnitude smaller than $1$, so the same type of fixed point multiplication can be used for all operations.