<p>You are given an integer <code>n</code>. You roll a fair 6-sided dice <code>n</code> times. Determine the total number of <strong>distinct</strong> sequences of rolls possible such that the following conditions are satisfied:</p>

<ol>
	<li>The <strong>greatest common divisor</strong> of any <strong>adjacent</strong> values in the sequence is equal to <code>1</code>.</li>
	<li>There is <strong>at least</strong> a gap of <code>2</code> rolls between <strong>equal</strong> valued rolls. More formally, if the value of the <code>i<sup>th</sup></code> roll is <strong>equal</strong> to the value of the <code>j<sup>th</sup></code> roll, then <code>abs(i - j) &gt; 2</code>.</li>
</ol>

<p>Return <em>the<strong> total number</strong> of distinct sequences possible</em>. Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>Two sequences are considered distinct if at least one element is different.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> n = 4
<strong>Output:</strong> 184
<strong>Explanation:</strong> Some of the possible sequences are (1, 2, 3, 4), (6, 1, 2, 3), (1, 2, 3, 1), etc.
Some invalid sequences are (1, 2, 1, 3), (1, 2, 3, 6).
(1, 2, 1, 3) is invalid since the first and third roll have an equal value and abs(1 - 3) = 2 (i and j are 1-indexed).
(1, 2, 3, 6) is invalid since the greatest common divisor of 3 and 6 = 3.
There are a total of 184 distinct sequences possible, so we return 184.</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 2
<strong>Output:</strong> 22
<strong>Explanation:</strong> Some of the possible sequences are (1, 2), (2, 1), (3, 2).
Some invalid sequences are (3, 6), (2, 4) since the greatest common divisor is not equal to 1.
There are a total of 22 distinct sequences possible, so we return 22.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= n &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution:
  def distinctSequences(self, n: int) -> int:
    mod = 1000000007
    faces = 6
    dp_range = 3
    
    def gcd(a: int, b: int) -> int:
      if a == 0:
        return b
      return gcd(b % a, a)
    
    dp = [[0 for _ in range(faces)] for _ in range(dp_range)]
    i1 = 0
    i2 = 1
    i3 = 2
    for face in range(1, faces + 1):
      dp[i1][face - 1] = 0
      dp[i2][face - 1] = 1
    
    for i in range(2, n + 1):
      for face in range(1, faces + 1):
        count = 0
        for p_face in range(1, faces + 1):
          if p_face == face or gcd(p_face, face) != 1:
            continue
          
          count += 1
          dp[i3][face - 1] = (dp[i3][face - 1] + dp[i2][p_face - 1]) % mod
        dp[i3][face - 1] = (mod + dp[i3][face - 1] - (count * dp[i1][face - 1] % mod)) % mod
        if i > 3:
          dp[i3][face - 1] = (dp[i3][face - 1] + dp[i1][face - 1]) % mod
      
      i1 = i2
      i2 = i3
      i3 = (i3 + 1) % dp_range
      dp[i3] = [0 for _ in range(faces)]
    #print(dp)
    ret = 0
    for face in range(faces):
      ret = (ret + dp[i2][face]) % mod
    return ret
          
    
    
421