How to prove a number is in the A002620 sequence.

Question

I am working on a project in PHP where I generate a number and want to see if said number is part of the A002620 sequence. I've started by looking at https://oeis.org/A002620 "Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4)."

I am confused how to implement, If I do floor(n^2/4) it proves nothing (below). I am a developer with basic Math so if someone could explain the process.

<?php
  $start_num = 9;
  $sum = floor($start_num**2/4);             
  echo $sum;
  //result 20
?>

Answer 1

At the given A002620 you find the following explicit formula for $a(n)$: $$a(n)={1\over8}\bigl(2n^2-1+(-1)^n\bigr)\qquad(n\geq0)\ .\tag{1}$$ This leads to the sequence $$0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, \ 90, 100 \ldots$$ When the number $a(n)=:a$ is given we immediately obtain from $(1)$ that $$n^2=4a+{1-(-1)^n\over2}\ .\tag{2}$$ Now it seems that you have arbitrary numbers $N\in{\mathbb N}$ and want to know whether this $N$ is an $a(n)$. From $(2)$ we can deduce the following:

• If $\sqrt{N}$ is integer then $N=a(n)$ for the even $n=2\sqrt{N}$.
• If $\sqrt{4N+1}$ is integer then $N=a(n)$ for the odd $n=\sqrt{4N+1}$.