连分数

连分数:

pkqk=a1+1a2+1a3+1a4+…=[a1,a2,a3,a4…] p k q k = a 1 + 1 a 2 + 1 a 3 + 1 a 4 + … = [ a 1 , a 2 , a 3 , a 4 … ]

可以得到以下关系：

⎧⎩⎨⎪⎪p1p2pk=a1=a1a2+1=akpk−1+pk−2(k=1)(k=2)(k>3) { p 1 = a 1 ( k = 1 ) p 2 = a 1 a 2 + 1 ( k = 2 ) p k = a k p k − 1 + p k − 2 ( k > 3 )

⎧⎩⎨⎪⎪q1q2qk=1=a2=akqk−1+qk−2(k=1)(k=2)(k>3) { q 1 = 1 ( k = 1 ) q 2 = a 2 ( k = 2 ) q k = a k q k − 1 + q k − 2 ( k > 3 )

大概可以这样写：

[pnqnpn−1qn−1]=[1001][a1110]…[an110] [ p n p n − 1 q n q n − 1 ] = [ 1 0 0 1 ] [ a 1 1 1 0 ] … [ a n 1 1 0 ]