How to count how many rails we need for a solar panel system

Hello, I have a complicated formula that I'm trying to create and can't figure out how to do it. I don't fully know how to even start this, but here's the first part. I want a formula where there are 3 numbers that are set. So the numbers are 3,4, and 5. I want to input a random number and what would populate would be a configuration of those 3 numbers to add up to the number inputted. So for example, if I put in the number 21, the configuration would be 5,4,4,4,4 (5+4+4+4+4). If the input number is 9, the output is 5,4 (5+4). The parameter would be 5, then 4, then 3. So for example for the number 21, you can also do 3 (7X), but the configuration is better with the number 5 rather than 3.


Now for some explanation. Essentially this formula is to count how many rails we need for a solar panel system. So each solar panel is about 66" X 39". Panels can be laid out in portrait (short distance 39") or in landscape (long distance 66") side by side. When the panels are laid side by side in a row it's called an array. So if a house had 8 modules in a row, then under 7 modules in a row and under that 3 modules in a row, that would be 3 arrays. One array of 8, one of 7 and one array of 3. Each row of modules have a gap of .4" between each module. Now the rails only come in 3 sizes. 132", 168" and 204". Each array needs 2 rows of rails under it. One row of rail at the top of the array and one row of rail at the bottom of it that goes all the way through the array. We splice the rails together to make longer rails than the 204" rails. At the end of the array there's 1" of rail on both ends. So essentially we are trying to get a formula that can tell us what size rails we need and how many of each size. We need have the least amount of waste. If the spliced rails come out longer than the 1" extension after the array, then the rail is cut. The minimum amount of rail that we could use if it's cut is 36". That's if the rail extends 36" from the previous array, then it's cut and we can use that for the next array to splice it. The system can be laid out all in portrait, it can be all in landscape or part portrait and part landscape.

Now back to the formula. When the system is laid out in portrait, the parameters are this. 132" rails are for 3 panels, 168" rails are for 4 panels, and 204" rails are for 5 panels. Again, each array requires 2 rails. So for 3 panels, it requires 2 rails of 132". So back to the original example. If there are 21 panels, the best configuration for the rails are 4 rails of 168" and 1 rail of 204". Multiply that by 2 for the top and bottom rails. If there are more arrays, then it's the same situation. If there's an array of 7 and an array of 2, then the configuration would be rails of 204" and rails of 168". Total rails, 2 rails of 204" and 2 rails of 168", one for the top and one for bottom.

When the system is laid out in landscape, the parameters are this. The parameters are based on length of the array which is the amount of panels, plus the gaps, plus the 1" on each side of the array. Instead of the simple 132" is 3 panels and so on.

When the system is laid out in portrait and landscape the parameters are first to try to keep the parameters of the portrait layout, but incorporate the landscape parameters for the landscape panels.



Please help me create this formula. It's a complicated formula, but even if only the first part can be made it will be very helpful.


Thank you!