Trigonometry and Polynomials?

Algebra Level 5

For every positive integer n,cos(nx)n, \cos (nx) can be written as a polynomial in cos(x)\cos (x) with integral coefficients. We call it Pn(y)P_{n}(y) where y=cos(x)y=\cos (x). For example, we have P1(y)=y,P2(y)=2y21P_{1}(y)=y, P_{2}(y)=2y^{2}-1.

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

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

S(n)S(n+A)(mod100). S(n) \equiv S(n+A) \pmod{100}.

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