During my work on sequencedb.net I bump into generated formulas that doesn’t exist in the Online Encyclopedia of Integer Sequences (oeis.org). Lately, I’ve started to submit them whenever I got some time over.
I recently proposed a formula for
A184517 Upper s-Wythoff sequence, where s=4n-2.
The discovered formula is simple
Where is the difference between the terms, to rewrite it without an explicit delta function we can rewrite it as
so we get
This is the formula I proposed after testing it against 10^6 first terms of A184517. An editor asked me if I could prove it which is a very good question since I hadn’t marked it as an empirical finding. Since the formula involves decimals, floor and ceiling it would certainly be comforting with a proof!
Looking in the program section of the entry, we can find the current formula used to generate the sequence, a bit simplified it can be written as
Now assuming this formula is correct, we need to prove that the new formula, .
Let’s start by simplifying by setting this is our connection to the Golden Ratio, .
Then we simplify it further, by removing some obvious cancellable terms
Focusing on whatever is inside the floor
With some further reworking we get to
Now we make a leap of faith and hope that whatever is inside the floor and ceiling is exactly apart
Recall that so on the lhs we get
Factoring out out gives us
Dividing it by two and multiplying the n part with 2 finally gives us
So we have proven that the expression inside the floor of minus 1 equals the ceiling of , hence they are the same.
I am sure this took far too many steps and detours before arriving at the conclusion - but I don’t want to spend too much time on this.
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