Arithmetic number puzzle

[pdurrant]

OK, after the last one, here's one that's purely arithmetic.

9, 14, 7, 11, 17, 26, 13, 20, 10, 5, ?, ?, ?, ?

What are the next four numbers in this sequence?
What are the rules to this sequence?
Why have I only asked for four more terms?

Don't forget to enclose answers in spoiler tags.

[dreams]

Yea! Another one! (of course, now I have to go somewhere this afternoon and can't spend time on it now) I look forward to trying this.

[obs20]

Originally Posted by pdurrant

OK, after the last one, here's one that's purely arithmetic.

9, 14, 7, 11, 17, 26, 13, 20, 10, 5, ?, ?, ?, ?

What are the next four numbers in this sequence?
What are the rules to this sequence?
Why have I only asked for four more terms?

Don't forget to enclose answers in spoiler tags.

Maybe this is it?

Spoiler:

8,4,2,1 after that it repeats 2,1,.....The rule: if the number is odd f(x)=(x+1)/2 if the number is even the next element is f(x)= x/2

[LazyScot]

Owww... Owwww. Brain hurts.

Thanks. I think. Cogitation required....

(And thanks for asking for use of spoiler tags.)

[pdurrant]

Originally Posted by obs20

Maybe this is it?

Spoiler:

8,4,2,1 after that it repeats 2,1,.....The rule if the number is odd x=x+(x+1)/2 if the number is even the next element is x/2

Yes, spot on. The interesting thing about the rules for this sequence is that it always seems to end up the same way no matter what number you start with. But — no-one has managed to prove that it works for all positive integers.

Although not quite the same, the rules for my sequence are effectively the same as for the sequence in the Collatz Conjecture, just missing out some terms.

http://en.wikipedia.org/wiki/Collatz_conjecture

But do see if you can work out the sequence yourself before 'cheating'.

[omk3]

I think it's:

Spoiler:

8,4,2,1 and from then on it's 2,1 for ever
if x/2 is an integer next number is x/2
if x/2 leaves a remainder, add result integer to remainder and all of it to x for next number.

It's late and I'm not sure I'm making sense, and I can't remember how to express this mathematically, so I analyse:
9/2=4, remainder 1. So next number is 9+4+1=14
14/2=7, so next number is 7
7/2=3, remainder 1. So next number is 7+3+1=11
....
5/2=2, remainder 1. So next number is 5+2+1=8
8/2=4, so next number is 4
4/2=2 so next number is 1
2/2=1,so next number is 1
1/2=0, remainder 1, so next number is 1+0+1=2, etc

(edit) The way obs20 put it is so much simpler... Result is the same though.

[pdurrant]

Originally Posted by omk3

(edit) The way obs20 put it is so much simpler... Result is the same though.

There are lots of ways to express it — yours worked, which is what really counts. Well done!

[LCF]

I would tip for the

Spoiler:

4,3,2,1

answer, but I have no idea why. But I'm working on it.