(note: P1 is the winner of RPS and P2 is the loser)
there are 3 types of stage lists for stage striking, according to the number of stages: 2n, 4n+1, and 4n+3.
I'll start by saying that, obviously, 2n is bad because it means that the players do not have the opportunity to strike an equal number of stages, therefore an obvious advantage comes up during striking
the difference between 4n+1 and 4n+3 is much less obvious, but it comes down to how the striking itself plays out. with 4n+1, it's trivial to achieve fair striking: P1 offers up the first piece of information in exchange for getting the final decision, whereas P2 sacrifices the influential final decision in favor of gaining some information from P1. this format is time tested, proven to work well, and is known to work fairly. the tradeoffs between P1 and P2 are close to even which is what makes this work. in particular, this can be seen in the null case where n=0. you have only a single starter, and there obviously is no advantage to either player while striking.
as for 4n+3, it's not as simple. how do you strike with say 7 stages? with 5 it's 1-2-1, with 9 it's 1-2-2-2-1 (1-2-1+1-2-1), etc. with 7 stages it's tougher. 2-3-1? 1-3-2? 1-2-2-1? 1-1-1-1-1-1? in each of these methods, striking is lopsided. for 2-3-1, the advantage is to P2, who gains twice as much information from P1 as he does with 5 starters, but still only sacrifices the influence of last ban, which time has demonstrated is roughly equivalent to one stage strike of information. for 1-3-2 the advantage is inverted - P1 offers up very little information and gets the extremely influential last pick out of 3 stages. this is far more valuable than picking from 2 stages and I hope is more clearly lopsided than 2-3-1. as for the other methods, 1-2-2-1 and 1-1-1-1-1-1, it's as simple as the fact that P1 is not trading the advantage of last pick for the disadvantage of first ban - P2 is left with first ban and P1 gets last pick. in the null case, n=0, this is accentuated very clearly: with 3 stages, P2 bans first and P1 picks. this was tried for a very short duration in smash 4 and was very clearly lopsided in favor of P1 before we returned to 5 starters.

**tl;dr: 4n+1 gives us the 1-2-1 banning structure which is the most fair we're aware of and is extendable to all lists of such a size, be it 1 5 9 or 41**