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Logic Level 5

Define a sequence of natural number X(n){X(n)} , where nn is any natural number, in which each digit of X(n)X(n) is either 00 or 11, by the following rules:

1.X(1)=11. X(1) = 1

2.2. We define X(n+1)X(n+1) as a natural number, which can be obtained by replacing the digits of X(n)X(n) with 11 if the digit is 00, and with 1010 if the digit is 11.

For example, X(1)=1X(1)=1, X(2)=10X(2)=10, X(3)=101X(3)=101, X(4)=10110X(4)=10110 and so on.

A(n)A(n) is defined as the number of digits in X(n)X(n).

B(n)B(n) is defined as the number of times 0101 appears in X(n)X(n).

For example, B(1)=0B(1)=0, B(2)=0B(2)=0, B(3)=1B(3)=1, B(4)=1B(4)=1, B(5)=3B(5)=3 and so on.

What is A(23)+B(23)A(23)+B(23)?

Note: There have been slight edits to the original question. The original question asks for a general formula of A(n)A(n) and B(n)B(n). Therefore, to provide for a numerical answer, the edits are necessary.

This algebra problem appears in the anime Puella Magi Madoka Magica. Students attending the Math class are just 14 years old, yet they are expected to solve the following question on the spot!

Translation credits: Puella Magi Wiki.

This problem is part of the question set Mathematics in Anime.

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