language agnostic - 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

Samples could be useful; the language is unimportant.

Answer 1

This touches an extremely deep, sophisticated and active area of mathematics. The solution is damn near trivial in some cases (linear recurrences) and damn near impossible in others (think 2, 3, 5, 7, 11, 13, ....) You could start by looking at generating functions for example and looking at Herb Wilf's incredible book (cf. page 1 (2e)) on the subject but that will only get you so far.

But I think your best bet is to give up, query Sloane's comprehensive Encyclopedia of Integer Sequences when you need to know the answer, and instead spend your time reading the opinions of one of the most eccentric personalities in this deep subject.

Anyone who tells you this problem is solvable is selling you snake oil (cf. page 118 of the Wilf book (2e).)