## Fibonacci Words

### Problem 230

Published on Saturday, 31st January 2009, 01:00 pm; Solved by 2040; Difficulty rating: 50%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

1415926535897932384689793238461415**9**265358979323846

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} 10^{n}× D_{A,B}((127+19`n`)×7^{n}) .