What are some algorithms for finding a closed form function given an integer sequence?

Question

I\'m looking form a programatic way to take an integer sequence and spit out a closed form function. Something like:

Given: 1,3,6,10,15

Return: n(n+1)/2

Answer 1

The Axiom computer algebra system includes a package for this purpose. You can read its documentation here.

Here's the output for your example sequence in FriCAS (a fork of Axiom):

(3) -> guess([1, 3, 6, 10, 15])

                 2
                n  + 3n + 2
(3)  [[function= -----------,order= 0]]
                     2
Type: List(Record(function: Expression(Integer),order: NonNegativeInteger))