4, 3/4, 4, 1/4, 1/12, 4, 1/36, 1/108, 1/324, 4, ... The next two terms are shown in green.
My guess is that the number 4 is in the positions a1, a3, a6, a10, a15, ... progressively moving further apart.
(Note that 1, 3, 6, 10, 15 also form a progression, being the sums of the natural numbers:
1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, ...)
The intervening terms form a geometric progression where the common ratio is ⅓:
3/4, 1/4, 1/12, 1/36, 1/108, ...