This sequence doesn't follow a simple arithmetic or geometric progression, but let's analyze the pattern.

Looking at the differences between successive terms:

- \( 4 - 5 = -1 \)
- \( 6 - 4 = 2 \)
- \( 15 - 6 = 9 \)
- \( 56 - 15 = 41 \)

The differences are \(-1\), \(2\), \(9\), and \(41\).

Now, let's look at the differences between these differences:

- \( 2 - (-1) = 3 \)
- \( 9 - 2 = 7 \)
- \( 41 - 9 = 32 \)

The second-level differences are \(3\), \(7\), and \(32\), which don't form a simple sequence either. However, it seems that each step involves a progressively more complex operation or multiplication.

Given the rapid increase in values, this suggests a possible exponential or factorial growth pattern mixed with addition.

One possible approach to predict the next term is to hypothesize that the pattern is a combination of previous terms with a multiplicative and additive component. We can estimate or experiment with potential formulas, but without a clear pattern, it remains speculative.

### Analyzing the progression:

If we try to apply a recursive relation, where each term is a function of one or more previous terms, we might consider higher-order polynomials or look at possible recursive multiplications.

Given the rapid increase (41 as a difference from 9), a plausible next difference could involve multiplication by around 4 or more.

Given the terms provided:
- \( 56 + \text{(estimated next difference)} \)
- If we assume the next difference could be in the range of 150-250 based on the observed exponential growth trend (e.g., multiplying 41 by a factor or adding a large number):

### Speculative Estimate:

Next number could be around \( 56 + 150 \) or more, so a potential candidate might be:

\[
\boxed{206}
\]

But this is highly speculative without a clear formula.

Would you like more details or have any questions about this approach?

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### Related Questions:

1. How do you identify different types of sequences?
2. What are common methods to solve complex sequences?
3. Can generating functions help in identifying sequence patterns?
4. What is the role of recursive formulas in sequences?
5. How do polynomials fit into sequence predictions?

### Tip:
When analyzing sequences, consider the possibility of a recursive or factorial pattern if simple arithmetic or geometric progressions don’t apply.

Explore the analysis of a complex sequence pattern involving exponential growth and factorial patterns. Understand the methodology used to predict the next term in a non-linear sequence. Suitable for advanced high school or college-level mathematics.

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