I came across with this cool Recurrence relation, an unfortunately i couldn't find sufficient mathematical tools to form it to a closed formula. i read several posts from math overflow saying that any linear recurrence can be made to a closed formula, but doesn't it even depends on the way the 'right wing elements' are chosen?

$a_n = \sum_{i=1}^{i<=n}a_{n-i}$
$a_0 = 1$ $a_1 = 1$ $a_2 =2$
where the index I run only on powers of 2.

for example the 7'th item by the recurrence relation is
$a_7 = a_6 + a_5 + a_3$

does those kinds of relations can be solved by a closed formula as well?