To find a formula for $U_n$ given the recurrence relation:

$U_{n+1} = \left( 3U_n + n^2 - n + 1 \right) \cdot 2^n$

Step-by-Step Solution:

1. Analyze the Relation:

• This is a recursive relation. To solve it, we would typically try to identify a closed-form formula or calculate specific terms if an initial value $U_0$ (or $U_1$) is provided.

2. Expanding Terms:

If $U_0$ or another initial value is given, calculate subsequent terms step by step to observe any patterns.

3. General Approach:

The recurrence involves two factors:

1. A multiplicative term $2^n$, suggesting exponential growth.
2. The term $3U_n + n^2 - n + 1$, which combines a linear recurrence and a polynomial.

This suggests that solving it analytically will likely involve:

• Homogeneous solution: Solve for $U_n$ assuming $n^2 - n + 1 = 0$.
• Particular solution: Solve for the polynomial and exponential contributions.

4. Numerical Calculation:

To find specific values for $U_n$, substitute the initial value and recursively calculate terms.

Questions for Clarification:

• What is the initial value $U_0$ (or $U_1$)?
• Is this a recurrence you want solved numerically or symbolically?
• Should we attempt approximation or exact computation of higher terms?
• Are there boundary conditions or constraints?
• Do you need a detailed explanation of methods for recurrence relations?

Tip:

For complex recurrences, breaking them into homogeneous and particular solutions simplifies the process significantly. Let me know if you'd like me to proceed with numerical calculations or a symbolic solution!