How do I even pronounce this word?

Like the Fibonacci sequence and Tribonacci sequence, define the Elevbonacci sequence such that its $$n^\text{th}$$ term, $$E_n$$ is the sum of the previous eleven terms with initial terms $$E_0 = E_1 = E_2 = \ldots = E_9 = 0, E_{10} = 1$$.

Define $$\displaystyle R = \lim_{n \to \infty} \frac { E_{n+1}}{E_n}$$.

Let $$f$$ denote the least degree monic polynomial with integer coefficients such that it has root $$R$$. Evaluate

$\sum_{f(r_{i}) = 0} r_{i}^{\deg(f)}$

where $$\deg(f)$$ is the degree of $$f$$.

Bonus: can you generalize this?

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