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

To solve this problem, we need to first understand the pattern and recursive nature of the snowflake construction. We can then use this understanding to calculate the values of $A(n)$ and $B(n)$ for each order $n$ snowflake. Finally, we can calculate the value of $G(n)$ for each order $n$ and find the sum of all $G(n)$ values from 3 to $10^7$.

Let’s start by analyzing the snowflake construction. We notice that for each order $n$ snowflake, we can obtain it by overlaying three identical order $n-1$ snowflakes. This means that the number of triangles in an order $n$ snowflake is three times the number of triangles in an order $n-1$ snowflake.

From this observation, we can define the following recursive formulas:
$$ A(n) = 3 \cdot A(n-1) $$
$$ B(n) = 3 \cdot B(n-1) + A(n-1) $$

We also know the base cases:
$$ A(1) = 1 $$
$$ B(1) = 0 $$

Now, let’s write the Python code to calculate $A(n)$ and $B(n)$ for any given value of $n$.

“`python
def snowflake_area(n):
if n == 1:
return 1, 0
else:
a_prev, b_prev = snowflake_area(n-1)
a = 3 * a_prev
b = 3 * b_prev + a_prev
return a, b
“`

With this function, we can calculate the values of $A(n)$ and $B(n)$ for any given $n$. Now let’s calculate $G(n)$ for each $n$ and find the sum $\sum_{n=3}^{10^7}G(n)$.

“`python
def gcd(a, b):
while b:
a, b = b, a % b
return a

def find_sum_g():
total_gcd = 0
for n in range(3, 10**7+1):
a, b = snowflake_area(n)
g = gcd(a, b)
total_gcd += g
return total_gcd
“`

Finally, let’s call the `find_sum_g()` function to get the desired sum.

“`python
sum_g = find_sum_g()
print(sum_g)
“`

Running this code will output the sum of $G(n)$ values from 3 to $10^7$.

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