For each Hecke group Z2*Z_k, the number of primitive reciprocal conjugacy classes of word length 2t grows like a constant times the t-th power of the dominant root of an explicit polynomial, and the same growth holds for primitive classes.
Buser, Geometry and spectra of compact Riemann surfaces
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Counting Reciprocal Hyperbolic Elements in Hecke Groups
For each Hecke group Z2*Z_k, the number of primitive reciprocal conjugacy classes of word length 2t grows like a constant times the t-th power of the dominant root of an explicit polynomial, and the same growth holds for primitive classes.