Well I don't see a pattern here
$$\{3,9,27,81,243,729,187,561,683,49,147,441,323,969,907,721,163,489,467,401,203,609,827,481,443,329,987\}$$

Created with *Mathematica* with the command

```
Table[If[k < 5, 3^k, FromDigits[IntegerDigits[3^k][[-3 ;; -1]]]], {k, 1, 27}]
```

In principle you calculate $3^k $ mod 1000, as 3 and 1000 are coprimes, there will be a pattern for sure, but the length of a period could be something up to 1000, which doesn't really help you.

Here the lenght of the period is 100, so the following numbers are recurring

```
3,9,27,81,243,729,187,561,683,49,147,441,323,969,907,721,163,489,467,401,203,609,827,481,
443,329,987,961,883,649,947,841,523,569,707,121,363,89,267,801,403,209,627,881,643,929,787,
361,83,249,747,241,723,169,507,521,563,689,67,201,603,809,427,281,843,529,587,761,283,849,
547,641,923,769,307,921,763,289,867,601,803,409,227,681,43,129,387,161,483,449,347,41,123,
369,107,321,963,889,667,1,3
```