First we need to set some binary functions:
Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let $\operatorname{wt}(n)$ be A000120, i.e., $1$'s-counting sequence: number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $\operatorname{val}(n)$ be A007814, i.e., exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
After that we need to set some functions based on Fibonacci numbers:
Let $f(n)$ be A000045, i.e., Fibonacci numbers: $f(n) = f(n-1) + f(n-2)$ with $f(0) = 0$ and $f(1) = 1$.
Let $g(n)$ be A010056, i.e., characteristic function of Fibonacci numbers: $a(n) = 1$ if $n$ is a Fibonacci number, otherwise $0$.
Let $h(n)$ be A072649, i.e., $n$ occurs $f(n)$ times.
Let $s(n)$ be a permutation of natural numbers, such that each $f(n)$ natural numbers are sorted in descending order. Here $$s(n)=f(h(n)+3)-n-1$$
Now we are ready to start:
Let $a_1(n)$ be the sequence of numbers $k$ such that $$\operatorname{wt}(k)=\operatorname{val}(k)+2$$
Let $$b_1(n)=\begin{cases} 2^{n-1},&\text{if $n<4$;}\\ 5,&\text{if $g(n-1)=1$;}\\ 2b_1(n-f(h(n-1)+1))-[g(n)=1],&\text{otherwise.} \end{cases}$$
Conjecture: $a_1(n)=a_1(n-1)+b_1(n)$ with $a_1(1)=3$.
Let $a_2(n)$ be the sequence of numbers $k$ such that $$\ell(k-1)-\operatorname{wt}(k-1)=\operatorname{val}(k)$$ with $a_2(1)=1$ prepended.
Let $$b_2(n)=\begin{cases} 2^{n-1},&\text{if $n<4$;}\\ 5,&\text{if $g(n)=1$;}\\ 2b_2(s(s(n)-f(h(n)+1)+1)+1)-[g(n-1)=1],&\text{otherwise.} \end{cases}$$
Conjecture: $a_2(n)=a_2(n-1)+b_2(n)$ with $a_2(1)=1$.
Is there a way to prove it?
