The next number in the series: $0$, $0$, $2$, $6$, $12$, $21$, $34$, $51$

I know that "next number" questions are not popular here, but this one is tough and I couldn't figure out the pattern. To give an idea one of the questions had a pattern of $$\operatorname{Prime}(\operatorname{Fib}(n))$$.

So this is the series:

$$0, 0, 2, 6, 12, 21, 34, 51$$ What is the next number in the series ?

• OEIS has nothing. Even when removing the initial $0$'s to account for possible different initial point. – Arthur Dec 5 '18 at 20:05
• Could you give some context? Where did you see this question? – Kevin Long Dec 5 '18 at 20:07
• My friend asked me this question, he said he saw it in a competition. – papabiceps Dec 5 '18 at 20:14
• @papabiceps Ew, a "what's the next number" question in a competition? Yuck. – Franklin Pezzuti Dyer Dec 5 '18 at 21:02
• The problem with "next number" questions is that unless a specific rule is given to generate the next number, it can be anything. That is why they are unpopular on Math.SE. +1 to Frpzzd's comment. – parsiad Sep 1 at 2:04

I'll find some pattern and then claim a correct answer:

0, 0, 2, 6, 12, 21, 34, 51

0, 2, 4, 6, 9, 13, 17

2, 2, 2, 3, 4, 4

Then 4 + 17 = 21 and 21 + 51 = 72 .

Well, {2, 2, 2, 3, 4, 4} calls for {2, 2, 2, 3, 4, 4, 4} but a continuing pattern is debatable after that. It might be {2, 2, 2, 3, 4, 4, 4, 5, ...}.

• Can you elaborate? – johnnycrab Dec 5 '18 at 20:53
• @johnnycrab Binomial transform. But not very convincing here. – Jean-Claude Arbaut Dec 5 '18 at 21:15
• @Jean-Claude Arbaut Thanks, didn't see it from just looking at the numbers. – johnnycrab Dec 5 '18 at 21:18