Counting Castles

We define a block to be a rectangle with a height of $1$ and an integer-valued length. Let a castle be a configuration of stacked blocks.
Given a game grid that is $w$ units wide and $h$ units tall, a castle is generated according to the following rules:
Blocks can be placed on top of other blocks as long as nothing sticks out past the edges or hangs out over open space.
All blocks are aligned/snapped to the grid.
Any two neighboring blocks on the same row have at least one unit of space between them.
The bottom row is occupied by a block of length $w$.
The maximum achieved height of the entire castle is exactly $h$.
The castle is made from an even number of blocks.
The following is a sample castle for $w=8$ and $h=5$:

Let $F(w,h)$ represent the number of valid castles, given grid parameters $w$ and $h$.
For example, $F(4,2) = 10$, $F(13,10) = 3729050610636$, $F(10,13) = 37959702514$, and $F(100,100) \bmod 1\,000\,000\,007 = 841913936$.
Find $(F(10^{12},100) + F(10000,10000) + F(100,10^{12})) \bmod 1\,000\,000\,007$.

To solve the problem, we need to determine the number of ways to express a given number as the sum of three triangular numbers.

Let’s start by creating a helper function to check if a number is triangular. A triangular number is a number that can be represented in the form of \( \frac{n(n+1)}{2} \), where \( n \) is a positive integer.

“`python
def is_triangular(num):
n = int((2 * num) ** 0.5)
return (n * (n + 1) // 2) == num
“`

Next, we can define a function called `count_triangular_sums` to calculate the number of ways to express a number as the sum of three triangular numbers.

“`python
def count_triangular_sums(num):
count = 0

for i in range(num + 1):
if is_triangular(i):
for j in range(i, num + 1):
if is_triangular(j):
for k in range(j, num + 1):
if is_triangular(k) and i + j + k == num:
count += 1

return count
“`

Now, we can use this function to find the value of \( G(17526 \times 10^9) \) by passing `17526 * 10**9` as the argument.

“`python
result = count_triangular_sums(17526 * 10**9)
print(result)
“`

Running this code will output the number of ways to express \( 17526 \times 10^9 \) as the sum of three triangular numbers.

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