Got an interesting J solution for the Podovan challenge; it's hardly golfy at all, but the method used is interesting. Th Podovan sequence is a linear recurrence, hence one can use that method with the characteristic polynomial of the recurrence relation.
The code below does this in the general case _if the polynomial has only simple roots_, which is the case here:
lr=: 2 : '1r2 <.@+ (n %. z ^~/~ i.#n) +/ . * (z=. >{:p. 1 ,~ -m) ^/ ]'