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This paper proposes a simplified algebraic model to present the complex configurations of single input change (SIC) circuits, and investigates the relationship between the SIC sequence and its generating circuit. Furthermore, several key properties of the studied sequence are provided and theoretically proved. Based on the above, we discover the commonness of different configurations of SIC circuits, and find the way to design the seed circuit at low overhead. To demonstrate our research results, a novel SIC circuit called seeded autonomous circular shift register (SACSR) circuit is presented. This new circuitry can automatically generate SIC sequences of more unique vectors. Finally, experimental results based on ISCAS85 Benchmark are provided to demonstrate the high performance of the proposed circuitry.