Since a connected graph with $n$ vertices and $n$ edges is unicyclic, you just need to subtract the ones whose cycle is a triangle (http://oeis.org/A053507) from the total (http://oeis.org/A057500).
Starting at $n=4$, I get 3, 72, 1500, 32280, 748440, 18898992, 520107840, 15555704400, 503580654720, 17569154733240, ... which is not in OEIS by itself.