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find the number of polynomial for a degree where f(x^2)=f(x)^2=f(f(x))

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TCS Other Question

Inspired by Fibonacci series, Sanket decided to create his own series which is 1,2,3,7,7,22,15,67,..... What number comes immediately before 63?
option
(a) 31
(b) 61
(c) 57
(d) 202
0,2,14,24,34,....?
And
2,15,41,80,132,....?
Expian and solve both...