how to find the $n^{th}$ term of the recurrence relation $a_n = a_{n - a_{n-1}} + a_{n-1 - a_{n-2}}$?

Question

I can only solve basic linear recurrence relations using characteristic polynomial technique. Then I hit this, and I cannot think of the polynomial that is necessary to solve this problem. If there is any concrete theory behind solving this kind of recurrence relations please provide source to them. Or if this can be solved by following some methods please illustrate how you would approach this kind of problems. The initial terms are $a_1= 1 , a_2 = 1.$

Answer 1

The sequence is given in OEIS A046699, where it is called the meta-Fibonacci sequence. It starts $$1, 1, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 8, 8, 8, 8, 9, 10, 10, \\ 11, 12, 12, 12, 13, 14, 14, 15, 16, 16, 16, 16, 16, 17, 18, 18, \\ 19, 20, 20, 20, 21, 22, 22, 23, 24, 24, 24, 24, 25, 26, 26, 27, \\ 28, 28, 28, 29, 30, 30, 31, 32, 32, 32, 32, 32, 32, 33, 34, 34, \\ 35, 36, 36, 36, 37 $$ It is remarked that $a_n=w(n-1)$ where $w(n)$ is the least $k$ such that $2^n$ divides $(2k)!$ After $1$, odd numbers appear once each and evens appear once more than the number of factors of $2$ in the number. I don't know how to prove that.