In the blog MRB Constant-D I noticed a peculiar outcome to several sets of equations involving f(n) = sin((a+b*floor(n))*Pi/M), where M is a constant to be explored, b is a number to be found and a is a "starting value" that causes f(n) ~=  -1, 0 or 1.

The peculiar outcome was to end up with a large set of approximations with respect to n that produce an f(n) near one of the integers {-1, 0, 1} -- by "harmonic oscillations" for consecutive integers, floor(n), where a ≠ b ≠ 0.  

 

I found several of these sets of equations in the blog that were true involving sin((a+b*floor(n))*Pi/m) where m = MRB constant.   Then at the end of the blog, I pretty much convinced myself that some similar sets exists for other constants.  

 

After my final I'm going to take up the study as to whether similar sets exists for all other constants, M, and how can I determine what b should be given  for each particular constant.   My work then should be mainstream enough where some other people might wish to contribute.