sympy and lru_cache

from sympy import sqrt, simplify
from functools import lru_cache

@lru_cache
def sinus_seq(n):
    if n == 0:
        return 0
    if n == 1:
        return (sqrt(6)-sqrt(2))/2
    return simplify((2*(sqrt(6)-sqrt(2))/4)*sinus_seq(n-1)-sinus_seq(n-2))



for i in range(200):
    print(f"{sinus_seq(i)}")