Is it possible to create a formula of a periodic sequence (for example: {1,2,3,1,2,3,...} or {0,1,0,-1,0,1,0,-1,...}), without the use of trygonometric functions (sin, cos...) and functions like floor.
#Formula of a periodic sequence without trygnometric functions and floor
inner quiver
idle sleetBOT
hard galleon
I have heard that the function y=A( (x/p)-⌊(x/p)⌋ ) is periodic without involving a trigonometric part,where A is the amplitude and p is the period.
(And the ⌊ ⌋ is the floor function.)
tranquil escarp
Periodicity is very related to roots of unity
You will likely not be able to never mention cos and sin if you study 3-periodic sequences
inner quiver
I found out it is possible to create a periodic sequence of {0,1,0,-1} in two ways.
It is also fairly easy to create a periodic sequence of two number for example 1 and 2.
But I'm still not able to figure out or find on the Internet how to create a periodic sequence of {1,2,3}.
tranquil escarp
For all sequences where the period is a power of 2, you can find formulas just the way you did
But for periods that are not, you will need roots of unity
A sequence is k periodic when its k-th iteration after rank n is equal to the term at rank n.
So $u_{n+3} = u_n$
If you look for a geometric sequence that satisfies this relation, that is $u_{n+1} = qU_n$ you get:
$q^3 = 1$
This gives you three possible values for q
shell saffronBOT
Daddy_314
scarlet junco
This also depends on what kind of formula you want. Can you use the mod function?
inner quiver
This also depends on what kind of formula you want. Can you use the mod function...
It says in the description that mod or floor isn't allowed
that would be just too easy
scarlet junco
It says in the description that mod or floor isn't allowed
You didn't write that part! 🤪
inner quiver
without the use of trygonometric functions (sin, cos...) and functions like floor.
I definitely wrote it at the beginning, but I suppose I've only noticed the message where I answered the first question.
scarlet junco
Using the mod function just circumvents the reality
I definitely agree, but I thought it was worth mentioning, partially because I think these questions aren't very well defined and it helps to show that.
scarlet junco
Oh, is mod a "function like floor"? Fair enough!
scarlet junco
No worries!
inner quiver
I had it in mind, but I haven't actually written it.
scarlet junco
There's some neat underlying math about how roots of unity and trig and modular arithmetic are connected.