We have a simple structure - biased rook of the two types.
Biased rook of the first kind which make open tours on a specific $f(n)\times 1$ board where $f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$ and cells are colored white or black according to the binary representation of $2n$. A cell is colored white if the binary digit is $0$ and a cell is colored black if the binary digit is $1$. A biased rook on a white cell moves only to the left and otherwise moves in any direction.
Let $T_1(n,k)$ is the number of open tours by a biased rook of the first kind on a specific $f(n)\times 1$ board, which end on a $k$-th cell from the right, where $f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$ with the same conditions for cells as shown above, then
$$T_1(n,k) = a_1(g(n) + 2^{k-2}(1 - L(n,k-2)))$$
for $1 < k \leqslant \left\lfloor\log_2{n}\right\rfloor$ + 1 with
$$T_1(n,1) = T_1(n, \left\lfloor\log_2{n}\right\rfloor + 2) = a_1(g(n))$$
where $$a_1(n)=(1+b_1(n))a_1(\left\lfloor\frac{n}{2}\right\rfloor), a_1(0)=1$$ $$b_1(n)=b_1(\left\lfloor\frac{n}{2}\right\rfloor)+n\bmod 2, b_1(0)=0$$ $$g(n)=n-2^{\left\lfloor\log_2{n}\right\rfloor}$$ $$L(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor \operatorname{mod} 2$$
Here $a_1(n)$ is the number of open tours by a biased rook of the first kind on a specific $f(n)\times 1$ board, which ends on any cell, where $f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$ with the same conditions for cells as shown above.
Biased rook of the second kind which make open tours on a specific $f(n)\times 1$ board where $f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$ and cells are colored white or black according to the binary representation of $2n$. A cell is colored white if the binary digit is $0$ and a cell is colored black if the binary digit is $1$. A biased rook on a white cell moves only to the left and otherwise moves moves only to the right.
Let $T_2(n,k)$ is the number of open tours by a biased rook of the second kind on a specific $f(n)\times 1$ board, which end on a $k$-th cell from the right, where $f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$ with the same conditions for cells as shown above, then
$$T_2(n,k) = a_2(g(n) - 2^{k-2}L(n,k-2)) + a_2(g(n) + 2^{k-2}(1 - L(n,k-2)))$$
for $1 < k \leqslant \left\lfloor\log_2{n}\right\rfloor$ + 1 with
$$T_2(n,1) = T_2(n, \left\lfloor\log_2{n}\right\rfloor + 2) = a_2(g(n))$$
where $$a_2(2n+1) = a_2(n), a_2(2n) = a_2(n) + a_2(n - 2^{b_2(n)}) + a_2(2n - 2^{b_2(n)}), a_2(0)=1$$ $$b_2(2n+1) = 0, b_2(2n) = b_2(n) + 1, b_2(1) = 0$$ $$g(n)=n-2^{\left\lfloor\log_2{n}\right\rfloor}$$ $$L(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor \operatorname{mod} 2$$
Here $a_2(n)$ is the number of open tours by a biased rook of the second kind on a specific $f(n)\times 1$ board, which ends on a white cell, where $f(n) = \left\lfloor\log_2{2n}\right\rfloor + 1$ with the same conditions for cells as shown above.
See also A284005, A329369, A329718.
Is there way to prove expressions for $T_1(n,k)$ and $T_2(n,k)$?
