I have a question regarding recurrence relations problem....If I have a hallway with length '1 by n' and 4 types of carpet: Red, Blue, Green, and Purple with sizes 1 by 5, 1 by 2, 1 by 1, and 1 by 1 respectively. There are some conditions as well, all green/purple carpets should appear AFTER red/blue carpets. Also, I cannot have green carpet AFTER purple. For example, an INCORRECT combination for a 1 by 10 hallway is 1 red carpet, 1 green carpet, 1 purple carpet, 1 green carpet, and 1 blue carpet is not allowed for the 2 reasons stated above. Because a green carpet is after a purple carpet and also the blue carpet is after green/purple. One CORRECT combination would be for the 1 by 10 hallway 1 red carpet, 1 blue carpet, 2 green carpets, and 1 purple carpet. I need to find the recurrent relation for this problem. I managed to solve most of it and came up with this so far: a_n= a_n-1 + (a_n-1 - a_n-2). However this recurrence relation does not take into account hallways that are made up of ONLY of RED and BLUE carpet. So the example I gave above would be counted in this recurrence relation because it has purple/green in the hallway. I am not sure how to count hallways with only Red/Blue carpet. Please explain how I can correct my recurrence relation to also count Red/Blue hallways. Here are some initial conditions: a₀ = 1, a₁ = 2, a₂=4, a₃=6, a₄=9,a₅=13, a₆=18. So to clarify a₆=a₅+(a₅-a₄)+(ONLY RED/BLUE HALLWAYS) Thanks.

Part 1: You are laying out carpet on a 1 meter by n meter hallway. You have available two different swatches of carpet: 1x5 red carpet and 1x2 blue carpet. Red Construct a recurrence with appropriate initial conditions for the number of ways to carpet a hallway of length n. - Justify your solution. Use as many initial condltions as you need, but no more. Blue Two different solutlons of length 12 No, can't overlap; this is bad No, can't go outside the lines; this is bad - No, can't leave gaps; this is bad Part 3: I I Same as part 1, except now we have green and purple swatches, each of which are 1x1. These can only appear Green Purple after all the red and blue swatches, and any greens must appear before any purples. No, can't have a blue swatch after green I purple No, can't have a green swatch after purple Three valid solutions of length 10