<p>Given an integer <code>n</code>, your task is to count how many strings of length <code>n</code> can be formed under the following rules:</p>

<ul>
	<li>Each character is a lower case vowel&nbsp;(<code>&#39;a&#39;</code>, <code>&#39;e&#39;</code>, <code>&#39;i&#39;</code>, <code>&#39;o&#39;</code>, <code>&#39;u&#39;</code>)</li>
	<li>Each vowel&nbsp;<code>&#39;a&#39;</code> may only be followed by an <code>&#39;e&#39;</code>.</li>
	<li>Each vowel&nbsp;<code>&#39;e&#39;</code> may only be followed by an <code>&#39;a&#39;</code>&nbsp;or an <code>&#39;i&#39;</code>.</li>
	<li>Each vowel&nbsp;<code>&#39;i&#39;</code> <strong>may not</strong> be followed by another <code>&#39;i&#39;</code>.</li>
	<li>Each vowel&nbsp;<code>&#39;o&#39;</code> may only be followed by an <code>&#39;i&#39;</code> or a&nbsp;<code>&#39;u&#39;</code>.</li>
	<li>Each vowel&nbsp;<code>&#39;u&#39;</code> may only be followed by an <code>&#39;a&#39;</code>.</li>
</ul>

<p>Since the answer&nbsp;may be too large,&nbsp;return it modulo&nbsp;<code>10^9 + 7</code>.</p>

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

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 5
<strong>Explanation:</strong> All possible strings are: &quot;a&quot;, &quot;e&quot;, &quot;i&quot; , &quot;o&quot; and &quot;u&quot;.
</pre>

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

<pre>
<strong>Input:</strong> n = 2
<strong>Output:</strong> 10
<strong>Explanation:</strong> All possible strings are: &quot;ae&quot;, &quot;ea&quot;, &quot;ei&quot;, &quot;ia&quot;, &quot;ie&quot;, &quot;io&quot;, &quot;iu&quot;, &quot;oi&quot;, &quot;ou&quot; and &quot;ua&quot;.
</pre>

<p><strong class="example">Example 3:&nbsp;</strong></p>

<pre>
<strong>Input:</strong> n = 5
<strong>Output:</strong> 68</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 2 * 10^4</code></li>
</ul>

MOD = 10**9+7
import numpy as np
def Matrix_power(A,n):
    res  = np.eye(5)
    while n>0:
        if n&1:
            res = res @ A
        A = A @ A
        n>>=1
    return res
class Solution:
    def countVowelPermutation(self, n: int) -> int:
        # a e i o u
        # 0 1 2 3 4
        # 1 02 0134 24 0
        
        @cache
        def dfs(i,s):
            res = 0
            if i==1:
                return 1
            if s==0:
                return dfs(i-1,1)+dfs(i-1,2)+dfs(i-1,4)
            if s==1:
                return dfs(i-1,0)+dfs(i-1,2)
            if s==2:
                return dfs(i-1,1)+dfs(i-1,3)
            if s==3:
                return dfs(i-1,2)
            if s==4:
                return dfs(i-1,2)+dfs(i-1,3)
            return res%MOD
        ans = sum(dfs(n,i) for i in range(5))%MOD
        dfs.cache_clear()
        # A = np.array([[0,1,1,0,1],[1,0,1,0,0],[0,1,0,1,0],[0,0,1,0,0],[0,0,1,1,0]])
        # P1 = np.array([[1],[1],[1],[1],[1]])
        
        return ans
        # A = np.array([[0,1,1,0,1],[1,0,1,0,0],[0,1,0,1,0],[0,0,1,0,0],[0,0,1,1,0]])
        # P1 = np.array([[1],[1],[1],[1],[1]])
        # PN = np.linalg.matrix_power(A,n-1) @ P1 
        # return int(sum(PN))%MOD
            
147817