Why does this ligand form 6-coordinate complexes with Transition elements and 8-coordinate complexes with Inner transition elements?
I decided to do some more research for this question as I had faced it before. Turns out the proper name of this ligand is diethylenetriaminepentaacetate ($\ce{DTPA^5-}$). It's an expanded version of EDTA and is primarily used in MRI scanning where it forms complexes with the Gadolinium ion. It forms complexes with a coordination number of six and seven with transition metals simply because these complexes are more stable. Here's Fe(III)-DTPA as an example1:

Experimentally Fe(III)-DTPA has been observed to adopt seven- and six-coordinate
crystal structures. This agrees with our result where the 8-coordinate Fe(III)-DTPA is $\pu{23 kJ mol^-1}$ higher in energy relative to the most stable 7-coordinate structure


With inner transition metals, it forms complexes with a denticity of 8, with Pr-DTPA being an example2:

References:

Sillanpää, Atte J., et al. “Density Functional Complexation Study of Metal Ions with (Amino) Polycarboxylic Acid Ligands.” Phys. Chem. Chem. Phys., vol. 5, no. 16, 2003, pp. 3382–93. doi:10.1039/B303234P.
Fusaro, Luca, et al. “Insight into the Dynamics of Lanthanide-DTPA Complexes As Revealed by Oxygen-17 NMR.” Inorganic Chemistry, vol. 51, no. 15, 2012, pp. 8455–61. doi:10.1021/ic3010085.