Problem 230

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).

斐波那契字

对于任意两个数字串A和B，我们定义F_A,B为序列(A,B,AB,BAB,ABBAB,…)，其中每一项都是拼接前两项而成。

进一步的，取F_A,B中首次包含至少n个数字的项，记其第n个数字为D_A,B(n)。

举例如下：

取A=1415926535，B=8979323846。我们希望找出D_A,B(35)。

F_A,B的前几项分别是：
1415926535
8979323846
14159265358979323846
897932384614159265358979323846
14159265358979323846897932384614159265358979323846

因此D_A,B(35)是第五项的第35个数字，也就是9。

现在我们取A是圆周率π小数点后的100位数字：

14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679

B是接下来的再100位数字：

82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196 .

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