What I was getting at is since the OP only brews once every 3 weeks, the yeast he saves may get old enough to the point where he would need to plate it to ensure a pure healthy pitch. The reliable shelf life of harvested rinsed yeast is 2 weeks (that's not to say that people can't and haven't made good beer with harvested yeast older than that). By brewing every 3 weeks, it would be impossible to maintain more than 1 harvested strain within the reliable shelf life. In order to get more reliable shelf life, he would need to store the yeast on a plate or slant (there are also longer term storage methods). That would require extra time and equipment that probably outweighs the cost benefit of reusing the yeast, or the payback time would be very long. Not to say that he shouldn't reuse yeast, but that he should stick with a single strain and rinse it between batches.
I would REALLY like to know where you get this '2 weeks' viability limitation on harvested/washed yeast... I think it's 100% weapons grade Bullonium. I actually have used yeast that was washed a month ago and it's going just fine after being pitched in just yesterday. I'm brewing about every two weeks, right now, in order to establish (and then maintain) a decent pipeline while giving each brew enough time to become excellent (3-4 week primaries for 'low' gravity brews, longer for higher).
From what I've read, you don't want to let yeast stick around for more than about 6 months in your fridge. I'm keeping it in the cooler area of the fridge, just above freezing (don't want to kill them wonderful/magical beasties)...
Keep in mind, I'm using smallish starters (about 1-1/2 quarts of starter) for these yeasts. I'm also harvesting just two or three doses per washing (lots of yeast per dose)... So, if you're thinking of harvesting a dozen doses from each, yeah, it's a lot to use. But, if you keep it within your production level, or pipeline, you can easily do it.
I do hope that my next place has enough room so that I can have more batches in process at a single time. Then I'll really want to make sure I wash yeast.
Also, right now I have two strains that I'm harvesting. Granted, the second strain doses have not been used, yet (two doses harvested) but I expect to brew next with one of them.
Thinking that you need to go to extremes to have viable yeast for brewing with 3 weeks between doesn't fly with me. Especially with what I've already encountered.
I think, that as long as you use the yeast within a reasonable amount of time, using a starter a day or two before your brew day, you'll be fine. There are many yeasts out there that are good for enough different styles (even if they don't list a style in their parameters) to make it work. I'm looking to have two to four strains on hand to brew with (washed)... These will be my main yeasts that will cover the majority of what I brew. Sure, there could be the odd time when I'll purchase a yeast strain for the infrequent brewing. I'm just looking to not purchase the same two or three strains for 90-95% of what I brew.
Another thing, that is probably an important item in the yeast bank... Labeling what strain they are, what generation they are, and the harvest/wash date will help you to ensure you don't overlook using a dose. Also keep the doses you harvest within your capacity for brewing. Don't have a dozen, or even half dozen, strains of yeast if you brew once a month. Also don't keep 6-8 doses of each yeast and expect to get through them all. Unless you're the yeast bank for a brew club that is.
Personally, I wouldn't mind sharing the yeast I've washed with another brewer, provided I get a dose to replace the one I give him/her when the brew comes off the yeast cake.
Properly store the washed yeast, and you should be good for some time with it. Or, develop a frozen bank of yeast if you're not going to use the strain for an extended period (more than 4 months). That could get a little pricey for initial (additional) hardware like a centrifuge (under $300 on amazon, with more than a few under $200)... Looking at it long term, that's not a lot to invest for a frozen yeast bank... Especially if you like some of the seasonal yeasts that are only available for 1-2 months a year.
Maybe I'm in the minority here, but I don't see anything wrong with anyone washing yeast that wishes to. Even if you only keep one or two doses going at a time, you'll end up saving cost...