Now armed with the power of ordinal numbers, the two players start playing on an infinite board. (Of course, no longer with real chocolates.) To be precise, the players are playing on a rectangular board \(n \times \omega\), where \(\omega\) is the smallest infinite ordinal. That is, the board consists of all lattice squares \((r,c)\) where \(0 \le r \le n-1\) and \(0 \le c\).
Among the games with \(1 \le n \le 100\), how many are won by the first player?