1026. Find the indices of all Fibonacci sequence members that are divisible by 3.
$\triangle$ Calculations show that the terms of the sequence divisible by 3 are those with indices 4, 8, and 12. It seems that this will always be the case if the index of the term is divisible by 4.

To prove this hypothesis, let's form a sequence of remainders from dividing the terms of the given sequence by 3:

$$
1 ; 1 ; 2 ; 0 ; 2 ; 2 ; 1 ; 0 ; 1 ; 1 ; 2 ; 0 ; 2 ; 2 ; 1 ; 0 ; \ldots .
$$

Here, a group of 8 remainders repeats periodically:

$$
1 ; 1 ; 2 ; 0 ; 2 ; 2 ; 1 ; 0
$$

The repetition is related to the fact that when adding $1+2$ or $2+1$, the sum 3 is replaced by 0, and the sum $2+2=4$ is replaced by 1 (since $4=3+1$).

Answer: all natural numbers divisible by 4.