MathDB

Let

a_0, a_1, a_2,\ldots

be a sequence of real numbers satisfying

a_0=1

and

a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}

for

n=1, 2,\ldots

Prove that there exists a positive integer

k

with

a_k<\frac{k}{2001!}

13