Integer points inside a square

[night berry]

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[toxic dawn]

if we start from the top of the square and count the number of lattice points in a row, pretty sure you get a pattern

[jagged jolt]

P_3 for your reference:

[jagged jolt]

Counted the half integers eyy?
I did that too

You don't really have to

The points in the upper half are in a 45-45-90 triangular configuration

You add 1 to 3 to 5 etc., which yields a perfect square

Same goes for the lower part

Let one half take the points on the x-axis, WLOG let the upper half do so

Then you have a sum of two perfect squares

All that remains is to express the bases of the perfect squares in terms of n

Let me walk you through P3. For the upper half, we have three layers of points:

• (0, 2); 1 point
• (-1, 1), (0, 1), (1, 1); three points
• (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0); five points
  for a total of 9 points, which is 3 squared

Similar counting yields 2 squared for the lower half

There are 5 points on the x-axis and in P_3 at the same time

That is one less than two times of 3, which is the base of the upper perfect square

[jagged jolt]

We count 5 since there are 2 to the left of the origin and 2 to the right

2 is in turn the greatest integer not exceeding half the diagonal length of the square, which is 3/sqrt(2)