To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the problem statement.

Let's define:
- $x_1$ as the number of heated seats sold.
- $x_2$ as the number of regular seats sold.

The objective is to maximize profit. Given that a profit of $30 is made on each heated seat and a profit of $20 is made on each regular seat, the objective function can be represented as:
\[ \text{Maximize: } 30x_1 + 20x_2 \]

Now, let's consider the constraints:
1. The arena can hold at most 300 people, so the total number of seats sold cannot exceed 300:
\[ x_1 + x_2 \leq 300 \]
2. At least 50 seats must be heated seats:
\[ x_1 \geq 50 \]
3. At least 3 times as many people prefer to sit in regular seats compared to heated seats:
\[ x_2 \geq 3x_1 \]

Given these definitions, the symbolic representation of the problem is:

```json
{
    'sym_variables': [('x1', 'number of heated seats sold'), ('x2', 'number of regular seats sold')],
    'objective_function': '30*x1 + 20*x2',
    'constraints': [
        'x1 + x2 <= 300',
        'x1 >= 50',
        'x2 >= 3*x1'
    ]
}
```

To solve this optimization problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Hockey_Arena_Profit")

# Define variables
x1 = m.addVar(name='heated_seats', vtype=GRB.INTEGER, lb=0)
x2 = m.addVar(name='regular_seats', vtype=GRB.INTEGER, lb=0)

# Set the objective function
m.setObjective(30*x1 + 20*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + x2 <= 300, name='total_capacity')
m.addConstr(x1 >= 50, name='min_heated_seats')
m.addConstr(x2 >= 3*x1, name='regular_vs_heated')

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Heated seats sold: {x1.x}")
    print(f"Regular seats sold: {x2.x}")
    print(f"Maximum profit: ${30*x1.x + 20*x2.x}")
else:
    print("No optimal solution found")
```