....What do they mean?! I couldn't help but think of that quote while doing this previous 236 assignment. That was still when I was doing the scratch work for the first question, trying to figure out all the right combinations of OMCOFS for all different lengths of binary strings.
At first, I only though of regular patterns like 00001111 and 00111100 and the like for a string of length 8, but there wasn't a clear pattern emerging from the numbers of H(2), H(3), H(4) and so on. Then I started getting creative. A string with four 1's and the rest zeros could still be an OMCOFS even if the four 1's weren't together, but still weren't odd when they were split. So I started thinking of combinations like 11001100 and 11011011. After using up 4 pages to come up with all the different, valid, types of binary combinations, I started to see a golden pattern emerging. H(1) = 1, H(2) = 2, H(3) = 3, H(4) = 5, H(6) = 8, and as Danny as told us before, H(0) = 1. Are you starting to see a pattern yet?
I was so ecstatic when I saw the fibonacci sequence, albeit offset by one index, that I couldn't believe it was that simple. So so I thought...coming up with a closed form for the fibonacci sequence was a challenge. I knew a form already existed, it could found with a simple Google search or a look through our notes, but I thought we might be expected to actually derive it by unwinding the recurrence, so that is what I set out to do. I unwound the recurrence 5 times, and instead of getting rid of recursive calls, more recursive calls to the function were being made, but in a peculiar manner. Every time you unwound the recurrence, the next level would be another form of a step down on Pascal's Pyramid. What were coefficients from his pyramid doing in this unwinding?
After a whole lot of research and brain racking, I ended up learning how to derive the golden ratio myself from a Fibonacci recurrence, and used that to find a slightly modified closed form for our recurrence, and well as prove that this closes form was equal to the recurrence. A lot of time was spent reading up in the golden ratio, how it is derived and what it represents, so even though it was a lot of time taken away from other courses, I do not regret it as I learned about something quite interesting.
I never thought I would ever say proof writing could ever get interesting...I think Danny might be changing my perception of this course and putting it in a new, good, light. I do not approve of it! I was so adamant to not like this theoretical course! I guess since I understand this time complexity stuff, this course is beginning to grow on me. But I still dislike induction. Especially structural. I am quiet anal about my likes and dislikes.
Also...I was told I am 5 episodes behind already for the new season of Dexter, and that just made me drop everything I was doing and start catching up. I was up until 3 last night watching the first two episodes, and I can't wait to get home and finish the last three! Do not fret, I won't be revealing anything in this post. I cannot promise the same about my next one though!
Is it weird that when I watch Dexter go about his.... "business" in the show, that I try to place myself in his shoes and try to imagine exactly how he's feeling? I try to imagine how I would react if that was me in Dexter's position, standing over the bound victim, getting ready to drive a blade through someone's heart. Does that make me psychotic or curious? Cause I swear I'm not[psychotic]! But hey...
The descent into hell is easy.
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