# A sequence of sorts

Moo Of Doom
1
2
1 2
4
1 1 3
6
1 2 4
?

What comes next?

Jimmy Snyder
Moo Of Doom said:
1
2
1 2
4
1 1 3
6
1 2 4
?

What comes next?

8 (a partition of 8, but the even numbers are partitioned into a single partition). I haven't figured out how the odd numbers are partitioned, but it seems that the entry for 2^n - 1 is 1 2 4 ... 2^(n-1)

Moo Of Doom
Nope, that's not it. But that's an interesting observation about the partitions! I hadn't noticed that.

ArielGenesis
8, i am not sure what the real pattern is but looking from the most left number in each line, 1,2,1,4,1,6,1,... one could easily guess 8

Dr. Robotnik
Worth a shot.

It increases the first number by one then multiply it with the rest.
ie:
1 2 4
(1+1) * 2 * 4 = 16

Learning Curve
Spoiler warning:

The answer is 16. Or so I think, if my way is allowed.

1
2
1 2
4
1 1 3
6
1 2 4
?

seperate the lines into groups just to understand it better:

1
2
--------
1 2
4
--------
1 1 3
6
------
1 2 4
?

Multiply the top lines digits by themselves and then by two:
(a*b)*2

so the first line:
(1)*2=2

and the second & third:
(1*2)*2=4

(1*1*3)*2=6

So for the bottom line you get:
(1*2*4)*2=16

ArielGenesis
so you are saying that the top line in each group generated by random ?

Moo Of Doom
Nope, none of the answers so far are correct. Jimmysnyder was the closest with his comment about the partitions.

Maybe there isn't quite enough information.

Here's the pattern a bit longer:

1
2
1 2
4
1 1 3
6
1 2 4
2 6
1 3 5
10
?

What comes next?

Jimmy Snyder

Moo Of Doom
Sure. Here's quite a bit more of the sequence:

1
2
1 2
4
1 1 3
6
1 2 4
2 6
1 3 5
10
1 1 1 2 6
12
1 5 7
?

Still want a clue? Partitions are correct. But think of the sequence as starting from 2 rather than 1...

0m3ga
a 16 year olds guess...lol

4 10
1 2 4 8
16

??

Last edited:
Moo Of Doom
Close. The second two are correct, but 4 10 is not. Care to explain your answer?

0m3ga
ummm lol nope - I am not good at explain them

ok its sumthin like 4 10...but the numbers add up to 14? i spent about 20 mins workin it out lol ill get it when i get time

Moo Of Doom
Yeah, the numbers add up to 14 but they're not 4 10.

0m3ga
is it 2 12?

0m3ga
and *this is gna seem really stupid* but what's partition?

Jimmy Snyder
0m3ga said:
and *this is gna seem really stupid* but what's partition?
A partition is simply a representation of a positive integer as a sum of other positive integers. For instance 12 = 2 + 10 or 12 = 4 + 8, or 12 = 1 + 2 + 9 or in the degenerate case, 12 = 12. The sequence that Moo has presented us with can be viewed as a sequence of partitions. Here is a list of all the possible partitions of 5:

5 = 1 + 1 + 1 + 1 + 1
5 = 1 + 1 + 1 + 2
5 = 1 + 1 + 3
5 = 1 + 4
5 = 1 + 2 + 2
5 = 2 + 3

I hope I didn't miss any. By the way, I didn't know what a partition was until I was about 21, so now you are 5 years ahead of me.

Moo Of Doom
0m3ga said:
is it 2 12?
Nope. First number is 2, though.

0m3ga
the only reasonable guess i can think off is 2 6 6

Moo Of Doom
Now you're just firing blindly :P

Nope, not 2 6 6.

0m3ga
lol yeah howd you guess

i see the whole pattern the only bit i didnt get was the bit where it went 2 6 lol and I am guessin that's linked to the line i can't get

gnpatterson
Looking at your pattern I think it has something to do with the factorisation of the next number

I can see that each number that is followed by a prime number is left intact

I think it is based on some method of decomposing the fractions

1/2, 2/3, 3/4, 4/5 etc

I notice that 2/3 can't be broken down
but 3/4 = 1/2 + 1/4
7/8 and 15/16 seem to match your breakdown

when I do 8/9=2/3+2/9 it seems to confirm your partition pattern

BUT

9/10=1/2 +4/10 DOESN'T match up

So I'm missing some other criterion that is not apparent

Homework Helper
Let's see here:

2 3 10
1 2 4 8
16
1 1 3 3 9
18

Moo Of Doom
NateTG has almost got it. His 2 3 10 doesn't add to 14, though... so that's not quite right. Everything else is correct though.

0m3ga
2 2 10?...

Moo Of Doom
0m3ga said:
2 2 10?...

Yes!

Now someone explain the pattern... :)

0m3ga
take any 2 subsequent even numbers from the sequence (e.g. 2 and 4) and the number between will = the smaller even number/the bigger even number

2
1 2
4

2/4=1/2

every odd number is written as a fraction where the last number os the denomenator and the sum of the rest equal the numerator e.g in the case of 1 1 1 2 6 = (1+1+1+2)/6

the other pieces i can't remembe right now lol someone else can finish it off

Homework Helper
The numbers are differences of the factors of n+1:

The factors of 15 are:
15,5,3,1
10=15-5
2=5-3
2=3-1

The factors of 16 are
16,8,4,2,1
8=16-8
4=8-4
2=4-2
1=2-1

The factors if 17 are
17,1
16=17-1

The factors of 18 are:
18,9,6,3,2,1
9=18-9
3=9-6
3=6-3
1=3-2
1=2-1

(It's pretty straightforward to show that this will always add up to n.)

Moo Of Doom
Congratulations, NateTG! 100% correct answer. That is the origin of the pattern indeed.