The sequence is defined as follows:

• $a_1 = 2$
• $a_2 = 3$
• For $k \geq 1$, $a_{2k+2} = 2 + a_k + a_{k+1}$ and $a_{2k+1} = 2 + 2a_k.$

We are tasked with determining all positive integers $n$ such that $a_n$ is an integer.

Step 1: Generate the sequence for the first few terms

Let's calculate some terms of the sequence to understand its behavior.

• $a_1 = 2$
• $a_2 = 3$

For $k = 1$:

• $a_3 = 2 + 2a_1 = 2 + 2 \times 2 = 6$
• $a_4 = 2 + a_1 + a_2 = 2 + 2 + 3 = 7$

For $k = 2$:

• $a_5 = 2 + 2a_2 = 2 + 2 \times 3 = 8$
• $a_6 = 2 + a_2 + a_3 = 2 + 3 + 6 = 11$

For $k = 3$:

• $a_7 = 2 + 2a_3 = 2 + 2 \times 6 = 14$
• $a_8 = 2 + a_3 + a_4 = 2 + 6 + 7 = 15$

For $k = 4$:

• $a_9 = 2 + 2a_4 = 2 + 2 \times 7 = 16$
• $a_{10} = 2 + a_4 + a_5 = 2 + 7 + 8 = 17$

Step 2: Identify the integers in the sequence

From the calculated values, we see that $a_1, a_2, a_3, \ldots$ are integers. This is not surprising, since the recurrence relations involve integer operations (addition and multiplication by 2), and the starting values are integers.

Thus, all terms of the sequence $a_n$ are integers for all $n$.

Step 3: Conclusion

Since all terms of the sequence $a_n$ are integers, every positive integer $n$ makes $a_n$ an integer.

Therefore, the answer is: All positive integers $n$ satisfy the condition.

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Would you like more details on the sequence behavior or the derivation of this conclusion? Here are some related questions you might find interesting:

1. How would the sequence behave if the initial conditions were different?
2. Can we prove this sequence formula by induction?
3. How could we compute the general form of the sequence term $a_n$?
4. What is the growth rate of this sequence as $n$ increases?
5. How does the recurrence relation change if the constants (e.g., 2) were different?

Tip: When working with recurrence relations, always start by calculating the first few terms to get an intuition about the sequence's behavior!