Oscillators with Least Transitions

[83bismuth38]

as the title says, the oscillators for as many periods as possible with the least amount of transitions.
b2e has a WHOOOOOOLE lot of them:
in this order: p2, 6, 14, 62, 126, 30:

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x = 53, y = 14, rule = B2e/S
39bo$40bo$26bo14bo$27bo14bo$15bo12bo14bo$16bo12bo14bo$17bo12bo14bo$18b
o12bo14bo$8bo10bo12bo14bo$9bo10bo12bo14bo$3bo6bo10bo12bo14bo$4bo6bo10b
o12bo14bo$o4bo6bo10bo12bo14bo$bo4bo6bo10bo12bo14bo!

interesting how it jumps from 126 to 30. I skipped one because it was also p14.
OTHER PERIODS:
p3 has alot:

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x = 3, y = 3, rule = B3ce/S012e
obo2$o!

[muzik]

The first example actually simulates a margolus neighborhood identical to the one simulated by 2x2, if you look closely. Same goes for B2c/S.

Also, if you look closely, you can see the periods are just twice mersenne numbers.

[dvgrn]

Also, if you look closely, you can see the periods are just twice mersenne numbers.

Not always, though.

The first exception is the eighteenth term in the series (OEIS A160657). The list and formula in the OEIS and in Nathaniel's paper are currently slightly inaccurate. The eighteenth oscillator ends up being period 524286/3=174762 instead of the full 524286 ticks.

If you can figure out exactly why those few oscillators end up with a shorter period, it might be a really new mathematical discovery. I think the next weird case is the 47th term. It's not just the numbers that are divisible by three, unfortunately; those show up in the list all over the place.

Weird and wonderful stuff --

[83bismuth38]

Weird and wonderful stuff --

that should be a famous quote in general.
anyways, i'm sad because lots of even numbered periods will be one transition, but p3 can't be. why can only a p3 not be phoenix? it really confuses me. also, p4:

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x = 5, y = 5, rule = B2e/S
bo$obo$bobo$2bobo$3bo!

p12:

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x = 9, y = 9, rule = B2e/S
bo$obo$bobo$2bobo$3bobo$4bobo$5bobo$6bobo$7bo!

this means more periods