A generalized triangle group is a group with a presentation

GT(l, m, n; w) = 〈a, b | al = bm = wn = 1〉;

where l, m, n [ges ] 2, and w is a cyclically reduced word in the free product on {a, b}. In the setting of one-relator products of cyclic groups, Fine et al. [3] calculated their Euler characteristics for n [ges ] 3 under certain conditions on the defining word w.