Question

What is the rule for this function?
input 1 output 2
input 2 output 5
Input 3 output 12
Input 4 output 34
Input 5 output 135
Input 6 output 741

Answer 1

e^x:
2.718
7.389
20.08
54.598
148.413
403.428
so, it grows faster than e^x as x gets large

maybe e^(kx^2) can be adjusted some. Since it is not at all obvious (to me), and since there are only 6 data points, it will be e possible to fit any suitable polynomial or exponential.

For example, Newton's divided difference method fits

f(x) = 2 + 3(x-1) + 2(x-1)(x-2) + (11/6)(x-1)(x-2)(x-3) + (6625/3000)(x-1)(x-2)(x-3)(x-4) + (103/40)(x-1)(x-2)(x-3)(x-4)(x-5)
= (1/120)(309x^5 - 4370x^4 + 23835x^3 - 61330x^2 + 73476x - 31680)