What Comes Next in This Number Sequence?

[Moo Of Doom]

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

What comes next?

 

[Jimmy Snyder]

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

What comes next?

Answer (in white):
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.
[Answer in white]

Answer: 16

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

 

[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
?

separate 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



Is that a logical answer?

 

[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, how about more clues, or an answer?

 

[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


??

 

[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]

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]

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

 

[NateTG]

Let's see here:

2 3 10
1 2 4 8
16
1 1 3 3 9
18
[/color]

 

[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]

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

 

[NateTG]

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.