# Which number comes next in this series?

The next numbers in this sequence:

1,1,3,6,12,22,39,67,113,188

What is the pattern?

• – user7530 Dec 5 '13 at 17:13
• It looks like A066982 on OEIS. – user113682 Dec 5 '13 at 17:19

After first 3 digits get sum or first 2 digits and add 3,4,5,6 For exampe (3+6)+3=12 (12+6)+4=22 (22+12)+5=39 (39+22)+6=67 (67+39)+7=113 and so on

The way you get to the next number in the sequence is to add the previous two numbers together, and add what term it is in the sequence (where we suppose it starts at the -1st term).

In other words, $a(n) = a(n-1) + a(n-2) + n$, where $a(-1) = a(0) = 1$ and start the recurrence.

So $a(1) = a(0) + a(-1) + 1 = 1 + 1 + 1 = 3$.

Similarly, $a(2) = a(1) + a(0) + 2 = 3 + 1 + 2 = 6$.

We do one more to guarantee the pattern: $a(3) = a(2) + a(1) + 3 = 6 + 3 + 3 = 12$, and so on.

Can you get the next number from this?

The numbers you wrote follow the rule $a_{0}=a_{1}=1$ (these are given), $a_{n}=a_{n-1}+a_{n-2}+(n-1)$ for $n\geq2$. Hence the number next to $a_{9}=188$ is $a_{10}=113+188+(10-1)=310$. It's a sort of Generalized Fibonacci Sequence.