\(-\)Trigonometry \(+\)and \(-\)Polynomials?

Algebra Level 5

This question is a follow up from Joel's problem regarding something very interesting I found.

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

Denote \(s(n)\) to be the sum of values of the coefficients of \({ P }_{ n }(y)\). For example, \(s(1)=1,\quad s(2)=1\)

Find the smallest positive integer value of \(A\) such that for all integers \(n>0\), \(s(n)\) and \(s(n+A)\) are congruent modulo \(1234\). Input \(A \times 3\) as your answer.

To avoid ambiguity, assume \(k>0\) if \(k\) is a positive integer.


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