How do I even pronounce this word?

Number Theory Level 5

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?

Inspirations: Problem 1 and Problem 2.

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