# Trigonometry and Polynomials?

Algebra Level 5

For every positive integer $$n, \cos (nx)$$ can be written as a polynomial in $$\cos (x)$$ with integral coefficients. We call it $$P_{n}(y)$$ where $$y=\cos (x)$$. For example, we have $$P_{1}(y)=y, P_{2}(y)=2y^{2}-1$$.

Denote $$S (n)$$ to be the sum of absolute values of the coefficients of $$P_{n}(y)$$. For example, $$S(1)=1, S (2)=3$$.

Find the smallest positive value of $$A$$ such that for all integers $$n> 0$$,

$S(n) \equiv S(n+A) \pmod{100}.$

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