To solve the problem, we analyze the relationship between (x) and (y) using the given data:

Step 1: Observe the pattern in (y) values

Given:
(x=1) → (y=3)
(x=2) → (y=6)
(x=3) → (y=11)
(x=4) → (y=18)
(x=5) → (y=27)
(x=6) → (y=38)

Step 2: Check differences

First differences (between consecutive (y)):
(6-3=3), (11-6=5), (18-11=7), (27-18=9), (38-27=11) (odd numbers increasing by 2).

Second differences:
(5-3=2), (7-5=2), (9-7=2), (11-9=2) (constant, indicating a quadratic relationship).

Step 3: Derive the formula

A quadratic equation is (y = ax^2 + bx + c). Since second difference is 2, (a = \frac{2}{2}=1).

Substitute (a=1) and (x=1):
(3 = 1(1)^2 + b(1) + c) → (b + c = 2).

For (x=2): (6 = 4 + 2b + c) → (2b + c = 2).

Solving: (b=0), (c=2).

Thus, the formula is (y = x^2 + 2).

Answer: (y = x^2 + 2) (or the next value if asked, e.g., (x=7) → (51)).

If the question expects the pattern expression, the answer is (\boxed{x^2 + 2}).

Assuming the next value is needed (common in such problems), for (x=7): (7^2 +2=51), so (\boxed{51}).

But based on typical pattern questions, the formula is the key. However, if the user expects the next number, it's 51. Given the context, likely the formula or next value. Let's go with the formula as the core answer:

(\boxed{x^2 + 2})

Or if numerical answer is needed (e.g., next term): (\boxed{51})

Since the original problem might have asked for the next term, let's confirm: the sequence after 38 is 51, so (\boxed{51}).

Final Answer: (\boxed{51})

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