题目230：斐波那契词语

欧拉计划

Fibonacci Words

For any two strings of digits, A and B, we define F_A,B to be the sequence (A,B,AB,BAB,ABBAB,...) in which each term is the concatenation of the previous two.

Further, we define D_A,B(n) to be the n^th digit in the first term of F_A,B that contains at least n digits.

Example:

Let A=1415926535, B=8979323846. We wish to find D_A,B(35), say.

The first few terms of F_A,B are:

1415926535
8979323846
14159265358979323846
897932384614159265358979323846
14159265358979323846897932384614159265358979323846

Then D_A,B(35) is the 35^th digit in the fifth term, which is 9.

Now we use for A the first 100 digits of π behind the decimal point:

14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679

and for B the next hundred digits:

82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196 .

Find ∑_{n = 0,1,...,17}   10ⁿ× D_A,B((127+19n)×7ⁿ) .

题目：

对于两个数字字符串 A 和 B，定义 F_A,B 为序列 (A,B,AB,BAB,ABBAB,...)，其中每一项为前两项的和。

进一步，我们定义 D_A,B(n) 为 F_A,B 中第一个至少含有 n 位数字的项的第 n 位数字。

例如:
设 A=1415926535，B=8979323846。我们希望求出 D_A,B(35)，如下所示。

F_A,B 的前几项为：

1415926535
8979323846
14159265358979323846
897932384614159265358979323846
14159265358979323846897932384614159265358979323846

因此 D_A,B(35) 为第五项的第 35 位数字，也就是 9。

现在我们设 A 为 π 的前 100 位小数：

14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679


并设 B 为接下来的 100 位小数：

82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196


求 ∑_{n = 0,1,...,17} 10ⁿ× D_A,B((127+19n)×7ⁿ)。