
    In a multi-zone emergency department triage system, determine the total number of patients that can be optimally distributed when operating at priority level n, where each level adds a triangular arrangement of patients to the previous level's distribution pattern. The calculation must account for the cumulative capacity across all levels up to and including the specified priority level, following a tetrahedral distribution pattern that ensures efficient resource allocation and patient flow.
    

    
def tetrahedral_number(n):
    """
    In a multi-zone emergency department triage system, determine the total number of patients that can be optimally distributed when operating at priority level n, where each level adds a triangular arrangement of patients to the previous level's distribution pattern. The calculation must account for the cumulative capacity across all levels up to and including the specified priority level, following a tetrahedral distribution pattern that ensures efficient resource allocation and patient flow.
    

    """
def tetrahedral_number(n):
    """
    In a multi-zone emergency department triage system, determine the total number of patients that can be optimally distributed when operating at priority level n, where each level adds a triangular arrangement of patients to the previous level's distribution pattern. The calculation must account for the cumulative capacity across all levels up to and including the specified priority level, following a tetrahedral distribution pattern that ensures efficient resource allocation and patient flow.
    

    """
    return n * (n + 1) * (n + 2) / 6