C. Palindromic Subsequences
brute force; constructive algorithms; math
For an integer sequence \(a = [a_1, a_2, \ldots, a_n]\), we define \(f(a)\) as the length of the longest subsequence\(^{\text{∗}}\) of \(a\) that is a palindrome\(^{\text{†}}\). Let \(g(a)\) represent the number of subsequences of length \(f(a)\) that are palindromes. In other words, \(g(a)\) counts the number of palindromic subsequences in \(a\) that have the maximum length.Given an integer \(n\), your task is to find any sequence \(a\) of \(n\) integers that satisfies the following conditions: \(1 \le a_i \le n\) for all \(1 \le i \le n\). \(g(a) > n\) It can be proven that such a sequence always exists under the given constraints.\(^{\text{∗}}\)A sequence \(x\) is a subsequence of a sequence \(y\) if \(x\) can be obtained from \(y\) by the deletion of several (possibly, zero or all) element from arbitrary positions. \(^{\text{†}}\)A palindrome is a sequence that reads the same from left to right as from right to left. For example, \([1, 2, 1, 3, 1, 2, 1]\), \([5, 5, 5, 5]\), and \([4, 3, 3, 4]\) are palindromes, while \([1, 2]\) and \([2, 3, 3, 3, 3]\) are not.
Each test contains multiple test cases. The first line contains the number of test cases \(t\) (\(1 \le t \le 100\)). The description of the test cases follows. The first line of each test case contains a single integer \(n\) (\(\color{red}{6} \le n \le 100\)) — the length of the sequence.Note that there are no constraints on the sum of \(n\) over all test cases.
For each test case, output \(n\) integers \(a_1, a_2, \ldots, a_n\), representing an array that satisfies the conditions.If there are multiple solutions, you may output any of them.
In the first example, one possible solution is \(a = [1, 1, 2, 3, 1, 2]\). In this case, \(f(a) = 3\) as the longest palindromic subsequence has length \(3\). There are \(7\) ways to choose a subsequence of length \(3\) that is a palindrome, as shown below: \([a_1, a_2, a_5] = [1, 1, 1]\) \([a_1, a_3, a_5] = [1, 2, 1]\) \([a_1, a_4, a_5] = [1, 3, 1]\) \([a_2, a_3, a_5] = [1, 2, 1]\) \([a_2, a_4, a_5] = [1, 3, 1]\) \([a_3, a_4, a_6] = [2, 3, 2]\) \([a_3, a_5, a_6] = [2, 1, 2]\) Therefore, \(g(a) = 7\), which is greater than \(n = 6\). Hence, \(a = [1, 1, 2, 3, 1, 2]\) is a valid solution.In the second example, one possible solution is \(a = [7, 3, 3, 7, 5, 3, 7, 7, 3]\). In this case, \(f(a) = 5\). There are \(24\) ways to choose a subsequence of length \(5\) that is a palindrome. Some examples are \([a_2, a_4, a_5, a_8, a_9] = [3, 7, 5, 7, 3]\) and \([a_1, a_4, a_6, a_7, a_8] = [7, 7, 3, 7, 7]\). Therefore, \(g(a) = 24\), which is greater than \(n = 9\). Hence, \(a = [7, 3, 3, 7, 5, 3, 7, 7, 3]\) is a valid solution.In the third example, \(f(a) = 7\) and \(g(a) = 190\), which is greater than \(n = 15\).