Generalized Fibonacci Recurrence

The recurrence belongs to the broader family of generalized Fibonacci sequences but has special combinatorial interpretations. All four branching sequences in the paper satisfy the same recurrence with different initial conditions. For coprime k and m, asymptotic behavior is expressed using the unique positive root of 1-x^k-x^m=0. The paper studies the recurrence g_n = g_{n-k}+g_{n-m} for positive integers k<m from branching and combinatorial perspectives.