Problem 570

Problem 570


A snowflake of order n is formed by overlaying an equilateral triangle (rotated by 180 degrees) onto each equilateral triangle of the same size in a snowflake of order n-1. A snowflake of order 1 is a single equilateral triangle.


Some areas of the snowflake are overlaid repeatedly. In the above picture, blue represents the areas that are one layer thick, red two layers thick, yellow three layers thick, and so on.

For an order n snowflake, let A(n) be the number of triangles that are one layer thick, and let B(n) be the number of triangles that are three layers thick. Define G(n) = gcd(A(n), B(n)).

E.g. A(3) = 30, B(3) = 6, G(3)=6
A(11) = 3027630, B(11) = 19862070, G(11) = 30

Further, G(500) = 186 and $\sum_{n=3}^{500}G(n)=5124$

Find $\displaystyle \sum_{n=3}^{10^7}G(n)$.





对于n阶雪花,记A(n)为厚度为一的三角形的数量,B(n)为厚度为三的三角形的数量。令G(n) = gcd(A(n), B(n))。

例如,A(3) = 30,B(3) = 6,G(3)=6
A(11) = 3027630,B(11) = 19862070,G(11) = 30

进一步地,G(500) = 186,而$\sum_{n=3}^{500}G(n)=5124$

求$\displaystyle \sum_{n=3}^{10^7}G(n)$。