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A cascade function is designed by combining two seed maps that resultantly has more parameters, high complexity, randomness, and more unpredictable behavior. In the paper, a cascade fractal function, i.e. cascade-PLMS is proposed by considering the phoenix and lambda fractal functions. The constructed cascade-PLMS exhibits the required fractal features such as fractional dimension, self-similar structure, and covering entire phase space by the data sequence in addition to the chaotic properties. Due to the chaotic behavior, the proposed function is utilized to generate a pseudo-random number sequence in both integer and binary format. This is the result of an extreme scalability feature of a fractal function that can be implemented on a large scale. A sequence generator is designed by performing the linear function operation to the real and imaginary part of a cascade-PLMS, cascade-PLJS separately, and the iteration number at which the cascade-PLJS converges to the fixed point. The performance analysis results show that the given method has a large keyspace, fast key generation speed, high key sensitivity, and strong randomness. Therefore, the scheme can be efficiently used further to design a secure cryptosystem with the ability to withstand various attacks.