# What will be the next term in this mathematical sequence?

What will be next in this series? $$0, 6, 24, 60, 120, 210, ...$$ I've tried it and noticed that the numbers are multiples of six. But I couldn't make a relation between them.

• the OEIS has three potential answers to this question. – Don Thousand Dec 25 '18 at 16:32

It seems like

sum of digit series
f(n)=n*sumOfDigits(n-1)*sumOfDigits(n+1)

example:-

f(1)=1*sumOfDigits(1-1)*sumOfDigits(1+1) = 1*0*2 = 0
f(2)=2*sumOfDigits(2-1)*sumOfDigits(2+1) = 2*1*3 = 6
f(3)=3*sumOfDigits(3-1)*sumOfDigits(3+1) = 3*2*4 = 24
f(4)=4*sumOfDigits(4-1)*sumOfDigits(4+1) = 4*3*5 = 60
f(5)=5*sumOfDigits(5-1)*sumOfDigits(5+1) = 5*4*6 = 120
f(6)=6*sumOfDigits(6-1)*sumOfDigits(6+1) = 6*5*7 = 210

f(7)=7*sumOfDigits(7-1)*sumOfDigits(7+1) = 7*6*8 = 336

• That's copying my solution – TheSimpliFire Dec 28 '18 at 13:27
• In my series the answer will change after 10th term, so how it's same. @TheSimpliFire – Vivek Kundariya Oct 24 '19 at 12:33

It seems like a

cube series.

Namely,

cube of 1 is 1 then 1 - 1 = 0
cube of 2 is 8 then 8 - 2 = 6
cube of 3 is 27 then 27 - 3 = 24
cube of 4 is 64 then 64 - 4 = 60
cube of 5 is 125 then 125 - 5 = 120
cube of 6 is 216 then 216 - 6 = 210
cube of 7 is 343 then 343 - 7 = 336

• Take differences between terms:

$$6, 18, 36, 60, 90, ...$$

• Notice that these are

$$6$$ times the triangular numbers $$1, 3, 6, 12, 15, ...$$

So the next difference should be

$$6\times 21 = 126$$

and the next term should be

$$210+126=336$$.

• The multiplication on the third spoiler is incorrect. And also, the sequence can be found on the OEIS as a nice formula: oeis.org/A007531 – athin Dec 25 '18 at 11:07

The $$n$$th term is $$n(n-1)(n+1)$$ and thus the required one is the seventh term giving an answer of $$7(6)(8)=336.$$