Assess students' ability to design a one-parameter family of nonlinear integer sequences (e.g., Somos-type recurrences) by proposing a recurrence dependent on an integer parameter k and integer initial seeds; require them to state necessary constraints on the seeds, prove that every term is an integer (using induction, divisibility invariants, Laurent phenomenon or combinatorial arguments), and derive either a closed-form formula for terms or the asymptotic growth rate (identify exponential/polynomial behavior, dominant roots, or use generating-function/tropical heuristics). Include one worked example for a specific k and justify each step rigorously.