I'm just curious why the formula for length
and the bandwidth changed as the antenna got
shorter.

Ken KG0WX

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The resonant length of an antenna depends on -

Length of wire.

But also to a far smaller extent on -

Increasing conductor diameter.
Increasing conductor insulation thickness.
Reducing height above ground.
Close proximity to antenna supports, buildings, trees and indoors.

All of which can reduce the resonant length to very slightly less than the
theoretical maximum value of 150/MHz metres for a halfwave dipole.

Or 492/MHz feet, pruned by a very few percent.

Behaviour versus the included angle of an inverted-V is slightly peculiar.
As the included angle approaches zero (which nobody ever uses) the resonant
length approaches that of an ordinary open-wire transmission line, 150/MHz
again.

"Very slightly less" is of the order of 1 or 2 or 3 percent unless you have
a VERY low antenna. Everybody's antenna is slightly different. Just keep a
pair of pruning shears handy. Or bend the wire back on itself.
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Reg, G4FGQ