A formula for any finite sequence of number In Advanced Problems in Mathematics by Stephen Siklos, pg24, he writes 

"Given any finite sequence of numbers, a formula can always be found
  which will fit all given numbers and which makes the next number
  (e.g.) 42."

Is there a source or proof for this statement?

Given a sequence of $a_0 ... a_{n-1}$, all you have to do is find $n$ linearly independent functions $f_0$ through $f_{n-1}$. Then define a sequence $c_i$ by $\sum_{i=0}^{n-1} c_i f_i(k)= a_k$ for all $k$. In matrix form, that's $M$c = a where $M$ is a matrix whose entries are given by $f_i(k)$ (the two parameters $i$ and $k$ giving a two dimensional array of numbers), c is the sequence of coefficients, and a is the original given sequence. If the columns of $M$ are linearly independent, then a solution for c exists, giving a formula for a.
One can obtain Lagrange polynomials from a special case where $f_i$ are powers of $k$, but one can use exponential functions, trigometric functions (e.g. discrete Fourier transforms), etc.
