Securing n Sheets with Thumbtacks: Can You Prove My Conjecture?”

[mrtwhs]

You have an infinite supply of square sheets of paper. You are going to secure these sheets on an infinitely large bulletin board by using thumbtacks. You must secure all four corners of each sheet however you may slightly overlap the sheets so that one thumbtack could secure up to four sheets at once. Under these assumptions, one sheet requires 4 tacks, 2 sheets require 6 tacks, 3 sheets require 8 tacks, etc.

What is the minimum number of thumbtacks needed to secure $$n$$ sheets?

I have a conjecture for a formula but have no clue how to prove it.

 

[mrtwhs]

Here is my conjecture for the number of thumbtacks needed for $$n$$ sheets of paper.

Spoiler

$$T(n) = \lceil(1+\sqrt{n})^2 \rceil$$ where the upper brackets represent the ceiling function.