What is the correct formula for Timer Balls?

Question

I was looking into the mechanics of Poké balls, and there's a lot of conflicting information about how the bonus from timer balls is calculated.

Earlier than Gen 5, everywhere agrees that the bonus starts at 1 and increases by 1 every 10 turns, to a maximum of 4 at t = 30.
Bulbapedia gives a continuous formula:

C = (10 + T) / 10 

but it's not clear whether that's supposed to be rounded down to the nearest integer.

From Gen 5 onwards, some places claim the calculation has changed to:

C = (1 + X * 1229/4096) 

However, the majority of sites only mention the other formula.

So are these two formulae correct for their respective generations? Are either of them rounded?

Answer 1

Generation III and IV

(("Amount of turns passed in battle" + 10) ÷ 10)x
[maximum 4×]

This means:

• 1× if less than 10 turns have taken place
• 2× if between 10 and 19 turns have taken place
• 3× if between 20 and 29 turns have taken place
• 4× if 30 or more turns have taken place

These are all rounded values.

As explained on Dragonflycave:

The Pokémon games prior to the fifth generation only work with integers. In practice, this essentially means that they round all numbers down, including at every intermediate step in a calculation.

Generation V and VI

(1 + "number of turns passed in battle" * 1229/4096)×
[maximum 4×]

This means you get approximately a 0.3 bonus each turn passed, up to 4x.

Other sources: Bulbapedia, Smogon, Thonky