Problem 1276

1. Define the structure of the isosceles trapezoid:
- The trapezoid consists of equal equilateral triangles with side length 1.
- The side $A_1E$ has length 3.
- The larger base $A_1A_n$ has length $n-1$.

2. Identify the layers:
- The trapezoid can be divided into layers, where each layer is a horizontal section between two lines.
- The bottom layer has $2n-2$ oblique segments.
- The middle layer has $2n-4$ oblique segments.
- The top layer has $2n-6$ oblique segments.

3. **Calculate the number of paths to point $E$:**
- To go from $A_1$ to $E$, we need to choose one oblique segment from each layer.
- The number of ways to choose these segments is:
$$(2n-2)(2n-4)(2n-6)$$
- Simplify the expression:
$$(2n-2)(2n-4)(2n-6) = 8n^3 - 48n^2 + 88n - 48$$

4. **Calculate the number of paths to point $\Delta$:**
- The number of paths to $\Delta$ is the same as to $E$:
$$8n^3 - 48n^2 + 88n - 48$$

5. **Calculate the number of paths to point $\Gamma$:**
- To go from $A_1$ to $\Gamma$, we only need to choose oblique segments from the bottom and middle layers.
- The number of ways to choose these segments is:
$$(2n-2)(2n-4)$$
- Simplify the expression:
$$(2n-2)(2n-4) = 4n^2 - 12n + 8$$

6. **Calculate the number of paths to point $B$:**
- To go from $A_1$ to $B$, we only need to choose oblique segments from the bottom layer.
- The number of ways to choose these segments is:
$$2n-2$$

7. Sum the total number of paths:
- The total number of paths is the sum of the paths to $E$, $\Delta$, $\Gamma$, and $B$:
$$2(8n^3 - 48n^2 + 88n - 48) + (4n^2 - 12n + 8) + (2n - 2)$$
- Simplify the expression:
$$2(8n^3 - 48n^2 + 88n - 48) + 4n^2 - 12n + 8 + 2n - 2 = 16n^3 - 96n^2 + 176n - 96 + 4n^2 - 12n + 8 + 2n - 2$$
$$= 16n^3 - 92n^2 + 166n - 90$$

The final answer is $\boxed{16n^3 - 92n^2 + 166n - 90}$