Equation for a sequence

[fredreload]

I want an equation generator for a set number of sequences, it's not an infinite sequence. For instance 71,73,70,56, I want an equation that would always give me a correct number nth number within this series or even for a series like this 1,1,1,1,2,1,1,1,1. It could be random sequences, but eventually I want to reduce this sequence to an equation so I can always find the nth number within this series, any number after this nth sequence doesn't matter. How can I do this?

[Sensei]

Sequence 1,1,1,1,2,1,1,1,1

assuming the first one is x=0

could be received with

 

[math]f(x)=2-sgn(abs(x-4))[/math]

 

https://en.wikipedia.org/wiki/Sign_function

x must be integer.

[fredreload]

Cool, I want an equation that would apply for any sequence, how about 71,73,70,56?

[Sensei]

Cool, I want an equation that would apply for any sequence,

[math]f(x)=sgn(abs(x-a_1))*b_1 + sgn(abs(x-a_2))*b_2 + .... + sgn(abs(x-a_n))*b_n[/math]

a_1....n are indexes.

b_1....n are values from sequence.

Edited September 4, 2016 by Sensei

[Sensei]

Cool, I want an equation that would apply for any sequence,

 

Corrected version:

[latex]f(x)=(1-sgn(abs(x-a_1)))*b_1 + (1-sgn(abs(x-a_2)))*b_2 + .... + (1-sgn(abs(x-a_n)))*b_n[/latex]

a_1....n are indexes.

b_1....n are values from sequence.

Edited September 5, 2016 by Sensei

[Klaynos]

A universal equation does not exist.

 

When in school I loved solving these kind of problems.

[fredreload]

Hmm this equation kind of grows though, but I get your point, I'm thinking of a single variable which doesn't seem possible

[Country Boy]

One of my teachers gave, as an example, the sequence 23, 25, 26, 18, 17, 16. Those are the numbers of the highway exits he passes on his way to work- he takes exit 26 to a second highway.

[wtf]

You can always fit finitely many values to a polynomial.

 

https://en.wikipedia.org/wiki/Lagrange_polynomial