Counting Castles

 Published on Saturday, 7th February 2015, 04:00 pm; Solved by 262;
Difficulty rating: 100%

Problem 502

We define a block to be a rectangle with a height of 1 and an integer-valued length. Let a castle be a configuration of stacked blocks.

Given a game grid that is w units wide and h units tall, a castle is generated according to the following rules:

  1. Blocks can be placed on top of other blocks as long as nothing sticks out past the edges or hangs out over open space.
  2. All blocks are aligned/snapped to the grid.
  3. Any two neighboring blocks on the same row have at least one unit of space between them.
  4. The bottom row is occupied by a block of length w.
  5. The maximum achieved height of the entire castle is exactly h.
  6. The castle is made from an even number of blocks.

The following is a sample castle for w=8 and h=5:


Let F(w,h) represent the number of valid castles, given grid parameters w and h.

For example, F(4,2) = 10, F(13,10) = 3729050610636, F(10,13) = 37959702514, and F(100,100) mod 1 000 000 007 = 841913936.

Find (F(1012,100) + F(10000,10000) + F(100,1012)) mod 1 000 000 007.