-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 nn, cos(nx)\cos{(nx)} can be written as a polynomial in cos(x)\cos{(x)} with integral coefficients. We would call it Pn(y){ P }_{ n }(y), where y=cos(x)y=\cos{(x)}. For example, we have P1(y)=y,P2(y)=2y21{ P }_{ 1 }(y)=y, { P }_{ 2 }(y)=2{ y }^{ 2 }-1

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

Find the smallest positive integer value of AA such that for all integers n>0n>0, s(n)s(n) and s(n+A)s(n+A) are congruent modulo 12341234. Input A×3A \times 3 as your answer.

To avoid ambiguity, assume k>0k>0 if kk is a positive integer.


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