Inductive proof for recursive formula with multiple recursive references

Question

This one is hard for me due to multiple recursive statements in the definition, and I have difficulty with inequalities.

$a_1=1$

$a_2=1$

$a_3=1$

$a_{n+3} = a_{n+2}+a_{n+1}+a_n$

Prove that for each natural number n with n>1: $a_n≤2^{n-2}$

Might be helpful for reference: $a_4=3, a_5=5,a_6=9, a_7=17$

When trying to prove P(n+1) I got: an+1=an+an−1+an−2=2n−2+2n−3+2n−4 which according to Wolfram Alpha is 7∗2n−4 though I don't know how that is reached. From there, I don't really know how to prove that

Answer 1

The base is obvious.

Let $a_k\leq2^{k-2}$ for all $3\leq k\leq n$.

Thus, $$a_{k+1}=a_k+a_{k-1}+a_{k-2}\leq2^{k-2}+2^{k-3}+2^{k-4}=7\cdot2^{k-4}<8\cdot2^{k-4}=2^{k-1}$$ and we are done!