Conjectured enumeration formulas for affine permutations avoiding patterns in $S_4$

Let $f_n^p$ denote the number of affine permutations of size $n$ that avoid the pattern $p$. The patterns under consideration are $3142$, $3412$, and $4123$.

Enumeration conjecture. The following equalities hold:

$$f^{3142}_n=\sum_{k=0}^{n-1}\frac{n-k}{n}\binom{n-1+k}{k}2^k$$

and

$$f^{3412}_n=f^{4123}_n=\frac{1}{3}\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{k}.$$

These formulas give the conjectured enumeration of affine pattern-avoiding permutations for the three remaining patterns in $S_4$; the source provides no proof or resolution beyond initial calculations.