data/problems/230.yml
---
:id: 230
:name: Fibonacci Words
:url: https://projecteuler.net/problem=230
:content: "For any two strings of digits, A and B, we define F<sub>A,B</sub> to be
the sequence (A,B,AB,BAB,ABBAB,...) in which each term is the concatenation of the
previous two.\n\nFurther, we define D<sub>A,B</sub>(<var>n</var>) to be the <var>n</var><sup>th</sup>
digit in the first term of F<sub>A,B</sub> that contains at least <var>n</var> digits.\n\nExample:\n\nLet
A=1415926535, B=8979323846. We wish to find D<sub>A,B</sub>(35), say.\n\nThe first
few terms of F<sub>A,B</sub> are: \n1415926535 \n8979323846 \n14159265358979323846
\ \n897932384614159265358979323846 \n1415926535897932384689793238461415 **9** 265358979323846\n\nThen
D<sub>A,B</sub>(35) is the 35<sup>th</sup> digit in the fifth term, which is 9.\n\nNow
we use for A the first 100 digits of π behind the decimal point:\n\n14159265358979323846264338327950288419716939937510
\ \n58209749445923078164062862089986280348253421170679\n\nand for B the next hundred
digits:\n\n82148086513282306647093844609550582231725359408128 \n48111745028410270193852110555964462294895493038196
.\n\nFind ∑<sub><var>n</var> = 0,1,...,17</sub> 10<sup><var>n</var></sup>×
D<sub>A,B</sub>((127+19<var>n</var>)×7<sup><var>n</var></sup>) .\n\n"