With the stuff I've seen in the literature of sequence transformations, I've started to love the formulae for Aitken's Δ² process:
$S_n^{\prime}=S_{n+1}-\frac{(\Delta S_n)^2}{\Delta^2 S_n}$
and its generalization the Wynn ε algorithm:
$\varepsilon_{k+1}^{(n)}=\varepsilon_{k-1}^{(n+1)}+\frac1{\varepsilon_{k}^{(n+1)}-\varepsilon_{k}^{(n)}}$
for the latter one especially because it is nicely represented as a lozenge diagram:
