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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.

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

Answer:-

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

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  • 2
    $\begingroup$ That's copying my solution $\endgroup$ – TheSimpliFire Dec 28 '18 at 13:27
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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

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  • 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

$210+126=336$.

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  • 2
    $\begingroup$ 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 $\endgroup$ – athin Dec 25 '18 at 11:07
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Simple Answer:

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

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