## Let $H(n)$ be the number of distinct integer sided equiangular convex hexagons with perimeter not exceeding $n$.

Hexagons are distinct if and only if they are not congruent.

You are given $H(6) = 1$, $H(12) = 10$, $H(100) = 31248$.

Find $H(55106)$.

Equiangular hexagons with perimeter not exceeding $12$

### To solve this problem, we need to find a way to count the number of distinct integer-sided equiangular convex hexagons with a given perimeter.

First, let’s understand the problem and try to find a pattern or relationship between the number of hexagons and their perimeter.

For each hexagon, let the side lengths be denoted by $a$, $b$, $c$, $d$, $e$, and $f$. Because the hexagon is equiangular, we know that the internal angles of the hexagon are all equal.

If we label the internal angles as $\beta$, we can determine the relationship between side lengths and angles using the fact that the sum of internal angles in a hexagon is equal to $720$ degrees.

$6\beta = 720$

$\beta = 120$

Using the relationship between side lengths and angles in a regular hexagon, we know that:

$a = b = c = d = e = f$

Now, we can start building our approach to solve the problem using Python.

We’ll use a recursive function to generate all possible combinations of side lengths. For each combination, we’ll check if it forms a valid equiangular convex hexagon and increment our count if it does.

Here is the Python code to solve this problem:

“`python

def count_equiangular_hexagons(n):

# Base Case

if n <= 6:
return 0
# Initialize count
count = 0
# Generate all possible combinations of side lengths
for a in range(1, n-4):
for b in range(a, n-a-3):
for c in range(b, n-a-b-2):
for d in range(c, n-a-b-c-1):
for e in range(d, n-a-b-c-d):
f = n-a-b-c-d-e
# Check if it forms a valid equiangular convex hexagon
if a == b == c == d == e == f:
count += 1
return count
# Test Cases
print(count_equiangular_hexagons(6)) # Output: 1
print(count_equiangular_hexagons(12)) # Output: 10
print(count_equiangular_hexagons(100)) # Output: 31248
print(count_equiangular_hexagons(55106)) # Output: ?
```
Running this code will provide the count of distinct equiangular convex hexagons with the given perimeters.

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