  In the context of extended Teleparallel gravity theories with a 3+1
dimensions Gauss-Bonnet analog term, we address the possibility of these
theories reproducing several well-known cosmological solutions. In particular
when applied to a Friedmann-Lema\^itre-Robertson-Walker geometry in
four-dimensional spacetime with standard fluids exclusively. We study different
types of gravitational Lagrangians and reconstruct solutions provided by
analytical expressions for either the cosmological scale factor or the Hubble
parameter. We also show that it is possible to find Lagrangians of this type
without a cosmological constant such that the behaviour of the LCDM model is
precisely mimicked. The new Lagrangians may also lead to other phenomenological
consequences opening up the possibility for new theories to compete directly
with other extensions of General Relativity.
