Let's analyze the sequence: $3, 2, 6, 6, 12, 18, 36, 56, \dots, 162$.

Step-by-Step Analysis:

We will look for patterns in the differences between consecutive terms.

1. First differences:

   • $2 - 3 = -1$
   • $6 - 2 = 4$
   • $6 - 6 = 0$
   • $12 - 6 = 6$
   • $18 - 12 = 6$
   • $36 - 18 = 18$
   • $56 - 36 = 20$

   First differences: $-1, 4, 0, 6, 6, 18, 20$

2. Second differences (differences of first differences):

   • $4 - (-1) = 5$
   • $0 - 4 = -4$
   • $6 - 0 = 6$
   • $6 - 6 = 0$
   • $18 - 6 = 12$
   • $20 - 18 = 2$

   Second differences: $5, -4, 6, 0, 12, 2$

The sequence does not follow a simple arithmetic or geometric progression, but it shows non-constant second differences. We need more information or a specific rule governing the sequence.

Would you like me to provide a conjecture based on the known terms or explore further?

Here are some related questions to consider:

1. Could the sequence involve alternating patterns?
2. What if we examine higher differences (third differences)?
3. Could this be part of a recursive sequence?
4. Is the sequence derived from a known mathematical series (e.g., Fibonacci-like)?
5. Should we check if there is a combination of addition and multiplication rules?

Tip: Sequences often have hidden patterns, and second or third differences can sometimes reveal them!