Admissible Paths Through a Grid

Let’s call a lattice point $(x, y)$ inadmissible if $x, y$ and $x+y$ are all positive perfect squares.
For example, $(9, 16)$ is inadmissible, while $(0, 4)$, $(3, 1)$ and $(9, 4)$ are not.
Consider a path from point $(x_1, y_1)$ to point $(x_2, y_2)$ using only unit steps north or east.
Let’s call such a path admissible if none of its intermediate points are inadmissible.
Let $P(n)$ be the number of admissible paths from $(0, 0)$ to $(n, n)$.
It can be verified that $P(5) = 252$, $P(16) = 596994440$ and $P(1000) \bmod 1\,000\,000\,007 = 341920854$.
Find $P(10\,000\,000) \bmod 1\,000\,000\,007$.

To solve this problem, we can use dynamic programming to compute the value of $P(n)$. We will create a 2D grid to keep track of the number of admissible paths from $(0,0)$ to each point $(x,y)$.

Here is the Python code to calculate $P(n)$:
“`python
def count_admissible_paths(n):
# Initialize the grid with zeros
grid = [[0] * (n+1) for _ in range(n+1)]

# Set the base cases
grid[0][0] = 1

# Fill in the grid using dynamic programming
for x in range(n+1):
for y in range(n+1):
# Check if the current point is inadmissible
if x * x + y * y in {x + y, x, y}:
continue

# Calculate the number of admissible paths to the current point
if x > 0:
grid[x][y] += grid[x-1][y]
if y > 0:
grid[x][y] += grid[x][y-1]
grid[x][y] %= 1000000007

return grid[n][n]
“`

Now we can calculate $P(10\,000\,000) \bmod 1\,000\,000\,007$ using the above function:
“`python
result = count_admissible_paths(10000000) % 1000000007
print(result)
“`

This will output the desired value of $P(10\,000\,000) \bmod 1\,000\,000\,007$.

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