Traditional named entity recognition (NER) aims to identify text mentions into pre-defined entity types. Continual Named Entity Recognition (CNER) is introduced since entity categories are continuously increasing in various real-world scenarios. However, existing continual learning (CL) methods for NER face challenges of catastrophic forgetting and semantic shift of non-entity type. In this paper, we propose GenCNER, a simple but effective Generative framework for CNER to mitigate the above drawbacks. Specifically, we skillfully convert the CNER task into sustained entity triplet sequence generation problem and utilize a powerful pre-trained seq2seq model to solve it. Additionally, we design a type-specific confidence-based pseudo labeling strategy along with knowledge distillation (KD) to preserve learned knowledge and alleviate the impact of label noise at the triplet level. Experimental results on two benchmark datasets show that our framework outperforms previous state-of-the-art methods in multiple CNER settings, and achieves the smallest gap compared with non-CL results.