So as I noted back in this post, I’ve got more fun things to share about this curious sequence. I got the idea from this video here, and decided to test it out on my own sequences. My “fibonacci grids” are similar to what is shown in the video, but in mine, you can see that at the beginning of each line, I retain the first number in my sequence on each line.
The basic premise of the video was that if one picks out a space of 2×2 numbers on this Fibonacci Grid, (he proceeded to write the original sequence and carry over no numbers, however) the determinant of the resulting “matrix” can be predicted cleverly by simply looking at the last number on the first line. What I’ve noticed is interesting because mine seems to rely on the parity of the starting number. My conjectures are as follows. I have not completed proofs as yet, but wanted to get this idea out there.
Conjecture 1. For the first and second values in a Fibonacci-like sequence (denoted here as . respectively) in a 4 by fibonacci square (using Nelson’s modified rules as noted in the original post), where , then the determinant of any 2 by 2 square is , where is the starting “row” of the fibonacci square.
Conjecture 2. For the first and second values in a Fibonacci-like sequence (denoted here as respectively) in a 4 by fibonacci square (using Nelson’s modified rules), where , then the determinant of any 2 by 2 square is , where is the starting “row” of the fibonacci square.
Now if only I were better at proofs…all the same, thought this was neat! Need to extend it to include arbitrary row lengths.