A shortened recurrence relation for the Bernoulli numbers

In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers $B_{2n}$, $n$ being any positive integer. This new recurrence seems advantageous in comparison to other known formulae since it allows the computation of both $B_{4n}$ and $B_{4n+2}$ from only $B_0,B_2,\mathrm{.}\mathrm{.}\mathrm{.},B_{2n}$.