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Hi Leandro.  No need to feel ashamed.  I concur that these equations don't all add up.  While I don't know how the initial author derived the 144k Ron value, it is safe to say that this is all in the ballpark.  Remember that this is a hysteretic device, so typically being in the ballpark is good enough for initial design.  The author tells us the following:

Vin = 24V +/- 10%

Vout = 11.8V (but by the earlier text, this number should be 3 * 3.9Vf + 0.2V = 11.9V)

fsw = 500kHz (this is just a target; the controller will move this)

An ideal/lossless buck will have D = Vo/Vi ~ 0.5 here

Since we know fsw and the duty cycle, we can determine ton ~1.0uS, and toff ~1.0uS.

While we don't know the exact values that the author may have used, it's safe to say that everything appears to be close to rough, back-of-the-envelope calculation.  It is possible to get to the 144k Ron number by modifying slightly Vi, Vo, fsw, or D.  If you derive D from the Ron equation in the Appendix, you get D=0.483.  Right there, that doesn't work for the nominal case as Vo/Vi would be higher than that, and we aren't taking into account any losses.  My guess is that some of this was derived experimentally, and the datasheet doesn't completely capture the design process.  For further insight into this part, I would recommend checking out application note, AN-1750.  It also might be helpful to put together a calculator with these equations to get a feel for how the variables play together in the system.  Sorry for the confusion.

Regards, Mike

Thanks Mike!