Let $M$ be the $n\times n$ matrix, known as the *GCD matrix*, of entries $M_{ij}=\gcd(i,j)$. In the paper

H J S Smith, *On the value of a certain arithmetical determinant*, Proc. London Math. Soc. **7**:208-212 (1875-76)

it is shown that $\det M=\prod_{k=1}^n\varphi(k)$; where $\varphi(k)$ is the Euler totient function.

Lucas introduced the family of sequences defined recursively by $L_0(s,t)=0, L_1(s,t)=1$ and
$$L_n(s,t)=s\cdot L_{n-1}(s,t)+t\cdot L_{n-2}(s,t).$$
**Remark.** My interest in these numbers lies in the fact that they are gcd-compatibile:
$$\gcd(L_i,L_j)=L_{\gcd(i,j)}.$$

Question 1.Can Smith's result be extended to the determinantal evaluation $$\det\left[\gcd(L_i(s,t),L_j(s,t))\right]_{i,j=1}^n \,\,?$$

**Update.** On the basis of Max Alekseyev's calculations depicted below, I state the following claim.

Conjecture.Define a modified Euler's totient function $\varphi_L(k):=\sum_{d\vert n}L_d(s,t)\cdot\mu\left(\frac{k}d\right)$. Then $$\det\left[\gcd(L_i(s,t),L_j(s,t))\right]_{i,j=1}^n=\prod_{k=1}^n\varphi_L(k).$$

**Remark.** Observe that $L_n(2,-1)=n$, therefore **Question 1** is already answered by Smith. In this case, $\varphi_L=\varphi$.

Question 2.We may start modest. If $s=2, t=1$ then $L_n(2,1)=P_n$ is the Pell sequence. What is the value of $$\det\left[\gcd(P_i,P_j)\right]_{i,j=1}^n\,\,?$$

Enumerative Combinatorics, vol. 1, 2nd ed., Exercise 3.96(a). $\endgroup$ – Richard Stanley Feb 14 '17 at 19:13strong divisibility sequences. In addition to the Fibonacci-type sequences defined by Lucas, another interesting class of strong divisibility sequences areelliptic divisibility sequences. $\endgroup$ – Joe Silverman Mar 1 '17 at 1:27