What is the equation of these coordinate points

Question

(2,2)
(3,6)
(4,24)
(5,120)
(6,720)
(7,5040)

I input all of these points into my calculator and ran every working equation none of which fits the equation. These equations include: Linear(obviously not), quadratic (also obviously not), cubic, quartic, natural log, exponential and power regressions none of which seem to fit. Any ideas?

Hint: Factorial function. Or else you can drag in the Gamma function. — André Nicolas, Sep 10 '14 at 21:36
I suspect that first point should be $(2,2)$; currently all pairs but the first are of the form $(n,n!)$. — Semiclassical, Sep 10 '14 at 21:39
There are 130 such sequences considering that the first term is $(2,2)$ but none if we let it be $(2,1)$, see OEIS. — Hakim, Sep 10 '14 at 21:59
There are infinitely many such sequences (though not necessarily interesting ones): Take the sequence to be points $(n, P(n))$ for $n = 2, \dots, 7$ for suitable polynomials $P$ of arbitrarily large degree. — anomaly, Sep 10 '14 at 22:21

Asimov · Answer 1 · 2014-09-10T22:02:47.103

1

This is the Gamma Function, the continuous version of the factorial. http://en.wikipedia.org/wiki/Factorial#The_Gamma_and_Pi_functions

edited Sep 10 '14 at 22:02

answered Sep 10 '14 at 21:36

Asimov

3,024

Why is the Gamma function necessary here? The arguments are integers, and all but the first are of the form $(n,n!).$ – Semiclassical Sep 10 '14 at 22:17
@Semiclassical He is looking for a function to connect these points. You need the Gamma function because it is continuous. – Asimov Sep 10 '14 at 22:18
Well, it depends on what the domain of the $x$-coordinate is. But I see your point, since the OP doesn't clarify that issue at all. – Semiclassical Sep 10 '14 at 22:25
So basically there is no equation where only x and y are involved, wow that's pretty interesting. I'd never even heard of "n!" thanks for response. – Diamond Louis XIV Sep 11 '14 at 02:25

What is the equation of these coordinate points

1 Answers1