Let's build a chocolate pyramid!CGAC2022 Day 1: Let's build a chocolate pyramid!

• Preferred date: Dec 1

I've got an infinite supply of two kinds of weirdly shaped chocolate:

• White chocolate, a square pyramid of side lengths 1
• Dark chocolate, a regular tetrahedon of side lengths 1

To celebrate the upcoming Christmas, I want to assemble them into a giant chocolate pyramid. When the base of the pyramid is a rectangle of size \$R \times C\$, the process to build such a pyramid is as follows:

1. Fill the floor with \$RC\$ copies of White chocolate.
2. Fill the gaps between White chocolate with Dark chocolate.
3. Fill the holes between Dark chocolate with White chocolate. Now the top face is a rectangle of size \$(R-1) \times (C-1)\$.
4. Repeat 1-3 until the top face has the area of 0.

The diagram below shows the process for \$2 \times 3\$. It takes 8 White and 7 Dark chocolate to complete the first floor, and 10 White and 8 Dark for the entire pyramid.

Given the width and height of the base rectangle, how many White and Dark chocolate do I need to form the chocolate pyramid?

You may assume the width and height are positive integers. You may output two numbers in any order.

Standard code-golf rules apply. The shortest code in bytes wins.

Test cases

(width, height) -> (white, dark)
(2, 3) -> (10, 8)
(10, 10) -> (670, 660)
(10, 1) -> (10, 9)