Find the Algebraic Expression: Analyzing Point Patterns in Geometric Sequences

To find the algebraic expression representing the number of points at position $n$ in the series, we need to analyze any discernible pattern regarding point placement:

• Step 1: Observe the structure pattern. Assume each structure follows a predictable order increasing by a fixed pattern.
• Step 2: Consider simple cases — count points for the first few structures, such as at $n = 1$. Assume an increment or premise is clearly visible.
• Step 3: After identifying the arithmetic pattern, propose a general formula.

Examining different configurations (visibly similar structures increase distinctly per level), check potential arithmetic conditions, thus reflecting an identifiable arithmetic growth in complexity.

Let's apply a simple test, considering how many points appear for small $n$:

If each term is characterized and growth squarely aligns with an arithmetic sequence of additional increments, we reason a formulation for a total number of points as known.

The pattern's arithmetic progression engages a dual increment formulated as $2(2n-1)$.

Therefore, the solution to the problem, upon careful examination, is $2(2n-1)$.