Problem 570

Snowflakes

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

雪花

将n-1阶雪花中每个等边三角形旋转180度，再覆盖到其本身上，就得到了一个n阶雪花；而最初的1阶雪花就是一个等边三角形。

雪花中的部分区域会被多次覆盖。在上图中，蓝色的区域的厚度为一，红色的区域为二，黄色的区域为三，依此类推。

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