Allow me grab this opportunity to suggest that whenever you could contribute to the OEIS to, well, do just that. If editing OEIS sequences appears tedious/daunting (in fact it is not), I'll offer every help I can. Best regards, jj * Andy Latto <andy.latto@pobox.com> [Apr 11. 2016 14:31]:
On Sun, Apr 10, 2016 at 8:28 PM, James Propp <jamespropp@gmail.com> wrote:
There's still the question of optimality. Last I checked, Andy Latto proposed an argument that I found unconvincing. Were other people convinced? If so, could someone explain to me why you should never jump a frog over a frog or a toad over a toad?
If you jump a frog over a frog, then you have two consecutive frogs, with the space behind them. any Toads that still need to pass these frogs will never be able to, so you had better only do this, if at all, at the very end, after all toads have passes all frogs. Since solutions can be time reversed, reversing the directions that toads and frogs move, the same argument shows that if you jump a frog over a frog, you had better do this only at the very beginning, before any frogs have jumped over any toads. (or to put it another way, if you jump a frog over a frog at the very end, after all the toads have passed over all the frogs, you can only do this from a position that is impossible to reach). So jumping frog over frog, or toad over toad, will always get you stuck.
(I hadn't included this in my argument, because I hadn't realized it was legal to jump a frog over a frog, but now see that it makes no difference whether it's legal).
Andy
Jim Propp [...]