# How is this number sequence built up and how does it continue?

Can you work out how this number sequence is built up? Can you work out how it continues?

1, 1, 2, 4, 8, 20, 60, 240, 1500, .......

Hint 1

this is a mathematical puzzle

Hint 2

It does not involve complicated mathematics

From time to time hints will be updated.

Note: this sequence cannot be found in The On-Line Encyclopedia of Integer Sequences

The next number is

16140

Following this pattern, given the sequence is $a_n$

$a_n = a_{n-1} + a_{n-2} \cdot (a_{n-3} + 1)$

Example: 60 = 20 + 8 * (4 + 1)

• Great job, well done. -- The solution you have is correct, but expressed ever so slightly differently to the way I had in mind, but it is equivalent. The way I put it together was $a_n = a_{n-1} + a_{n-2} + a_{n-2}.a_{n-3}$. So Well done, congratulations.
– tom
Mar 7, 2018 at 10:01

It seems like I was way off on the pattern I was trying to track. Seems like this is a neat coincidence though.

• What does this mean? I'm confused. Mar 6, 2018 at 21:52
• looks interesting, but I don't understand this either - can you explain a bit more?
– tom
Mar 7, 2018 at 9:58
• It's the distances between the first point and all the other points in the sequence plotted as (x,y) where x = 1, 2, 3. . . and y = the sequence. It seems like distance = y(rounded to the closest even number) Mar 7, 2018 at 14:28
• What you effectively have shown there is that $$Lim_{n \to \infty} \sqrt{(X_n - 1)^2 + (Y_n - 1)^2} \to Y_n$$. Which is another way of saying Y grows much faster than x... or by more than 1 per term. Mar 7, 2018 at 14:59

Obligatory nth order polynomial:

$$f(x) = \frac{5}{384}x^8-\frac{4597}{10080}x^7+\frac{19471}{2880}x^6 - \frac{19831}{360}x^5+\frac{308935}{1152}x^4-\frac{1141669}{1440}x^3+\frac{1982957}{1440}x^2-\frac{354937}{280}+465$$

Predicted next five values:

8116, 33619, 112289, 318913, 800021

• I dislike people reflexively pointing out that there is always an nth-order polynomial (or spline) that fits n arbitrary points. Anyway the spirit of this question forbade it: "It does not involve complicated mathematics" (yeah you could try to argue that nth-order interpolation doesn't). If the OP wanted to be more lawyerly they could have said "the answer has < n characters and only requires the four elementary operations".
– smci
Mar 7, 2018 at 3:34
• "If one plugs it into the given numbers, they all match... the next fives values are [given integers]" I doubt it does, it looks like you're doing rounding, most of your subterms aren't even integers.
– smci
Mar 7, 2018 at 3:36
• Ugh, freaking datatypes. I guess I should've done it by hand. Anyways, I've cleaned up the post after realizing how abrasive it was. Mar 7, 2018 at 4:17
• Your post was perfectly fine, just without the first sentence, also noting that in general that f(n) will give non-integer values. Curious which language you used, Python or what?
– smci
Mar 7, 2018 at 4:40
• Interesting solution - sorry not correct, but +1 for the polynomial fitting
– tom
Mar 7, 2018 at 9:58