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# 矩阵快速幂 优化
mod = 10**9+7
class MatPow:
def __init__(self,A): # k 阶递推式需要给定前 k*2 项
k = len(A)//2
self.f = A[:k]
self.A = A
self.g = self.gen(A)[::-1]
def gen(self,A): # Berlekamp-Massey 算法，给定前 k*2 项 A，返回符合的最短系数组 g
pre_c = []
pre_i, pre_d = -1, 0
g = []
for i,a in enumerate(A):
d = (a-sum(x*A[i-1-j] for j,x in enumerate(g)))%mod
if d == 0:
continue
if pre_i<0: # 首次算错，初始化 g 为 i+1 个 0
g = [0]*(i+1)
pre_i,pre_d = i,d
continue
bias = i-pre_i
old_len = len(g)
new_len = bias + len(pre_c)
if new_len>old_len: # 递推式变长了
tmp = g[:]
g += [0]*(new_len-old_len)
delta = d*pow(pre_d,-1,mod)%mod
g[bias-1] = (g[bias-1]+delta)%mod
for j,c in enumerate(pre_c):
g[bias+j] = (g[bias+j]-delta*c)%mod
if new_len>old_len:
pre_c = tmp
pre_i,pre_d = i,d
return g
def get(self,n): # Kitamasa 算法，给定前 k 项 f 和系数组 g，求第 n 项
def compose(A,B): # 根据 g(n) 的系数组 A 和 g(m) 的系数组 B 计算 g(n+m) 的系数组
C = [0]*k
for a in A:
for j,b in enumerate(B):
C[j] = (C[j]+a*b)%mod
B = [((B[i-1] if i else 0)+B[-1]*g[i])%mod for i in range(k)]
return C
f,g = self.f,self.g
if n<len(f):
return f[n]%mod
k = len(g)
if k == 0:
return 0
if k == 1:
return f[0]*pow(g[0],n,mod)%mod
res = [0]*k
C = [0]*k
res[0] = C[1] = 1
while n:
res = compose(C,res) if n&1 else res
C = compose(C,C)
n >>= 1
return sum(a*b for a,b in zip(res,f))%mod
class Solution:
def countVowelPermutation(self, n: int) -> int:
A = [0,5]
a,e,i,o,u = 1,1,1,1,1
for _ in range(10):
a,e,i,o,u = (e+i+u)%mod,(a+i)%mod,(e+o)%mod,i,(i+o)%mod
A.append((a+e+i+o+u)%mod)
mp = MatPow(A)
return mp.get(n)
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