Let $R$ be a commutative ring with unit, local and Noetherain, with $\mathcal{M}$ maximal ideal. Let $ K $ be the residue field. Define $$ \phi(n) := dim_{K} \frac{\mathcal{M}^n}{ \mathcal{M}^{n+1} } $$ If $ \phi (0)=1, \phi (1)=3, \phi (2)=5 $, show that $ \phi (3) \leq 7 $.
I tried by Nakayama to lift the basis of the different quotients to get a minimal set of each $ \mathcal{M}^{k} $, $k=1, 2 $, but this did not get me anywhere. I guess that one has to work on generating sets directly, but all the bounds I get are not that sharp.
Thank you very much