Question 18
A sequence of polynomials $a_n(x)$ is defined recursively by
$$a_0(x)=1,\qquad a_1(x)=x^2+x+1,$$
$$a_n(x)=(x^n+1)a_{n-1}(x)-a_{n-2}(x)\quad(n\ge2).$$
For example,
$$a_2(x)=(x^2+1)(x^2+x+1)-1=x^4+x^3+2x^2+x,$$
$$a_3(x)=(x^3+1)(x^4+x^3+2x^2+x)-(x^2+x+1)$$
$$=x^7+x^6+2x^5+2x^4+x^3+x^2-1.$$
Evaluate $a_{2010}(1)$.
$$a_0(x)=1,\qquad a_1(x)=x^2+x+1,$$
$$a_n(x)=(x^n+1)a_{n-1}(x)-a_{n-2}(x)\quad(n\ge2).$$
For example,
$$a_2(x)=(x^2+1)(x^2+x+1)-1=x^4+x^3+2x^2+x,$$
$$a_3(x)=(x^3+1)(x^4+x^3+2x^2+x)-(x^2+x+1)$$
$$=x^7+x^6+2x^5+2x^4+x^3+x^2-1.$$
Evaluate $a_{2010}(1)$.