# These sums are getting out of hand

Discrete Mathematics Level 5

Let $$\mathbb{Z}^*$$ be the set of non-negative integers.

Given that $$\lambda_0, \lambda_1, \lambda_2, \ldots$$ are real numbers that satisfy, for all $$n \in \mathbb{Z}^*$$,

$\displaystyle \sum_{a+b+c=n} \lambda_a \lambda_b \lambda_c = 1$

where the sum is taken over all $$a,b,c \in\mathbb{Z}^*$$ such that $$a+b+c = n$$. (In particular, $$\lambda_0 = 1$$.)

It turns out that for all $$n \in \mathbb{Z}^*$$ there is a nice closed form for $$\lambda_n$$. Using such a form, determine $$\lfloor 1000 \lambda_{2016}\rfloor$$.

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