sequence question

nha · 2010-09-11 23:34:47

Hey everyone, I have a simple question.

How can I find a formula for the following sequences (like in the form ((n+1)n)/2 etc):

0,0,1,1,2,4,7,13,24

and

2,1,3,4,7,11,19,30

Thanks

bobbym · 2010-09-12 04:38:14

Hi nha;

For the first, since it looks like you are interested in an interpolating polynomial:

or

May look a little strange because it is hornerized for efficient computation.

soroban · 2010-09-12 15:59:49

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. . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . .

But check my work . . . please1
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nha · 2010-09-12 16:19:14

Thanks bobbym and soroban. I feel stupid now, I didn't know how complicated the functions were. I really just want to find a pattern so I can find the next 2 numbers in each sequence.

I got it for the first:
0+0+1=1
0+1+1=2
1+1+2=4
1+2+4=7
2+4+7=13
4+7+13=24

7+13+24=44
13+24+44=81

But the second I can't seem to get. That 19 is my issue.

Thanks again, and sorry for having you finding the function, I didn't think they would be so big and difficult.

bobbym · 2010-09-12 16:25:55

Hi nha;

And it gets worse. There are an infinite amount of functions that can go through those set of points. Picking the next two numbers is actually mathematically impossible. Even a set that looks like this { 1,2,3,4,5,6,7,...}, humans guess 8 and because the questioner is also a human they are marked right. Actually the next number could be -12364 or anything else. Point is I could fit a different function to those points that would always produce a different 8th number. Still we try to guess at the simplest form that the questioner meant.

nha · 2010-09-12 16:34:01

bobbym wrote:

Hi nha;
And it gets worse. There are an infinite amount of functions that can go through those set of points. Picking the next two numbers is actually mathematically impossible. Even a set that looks like this { 1,2,3,4,5,6,7,...}, humans guess 8 and because the questioner is also a human they are marked right. Actually the next number could be -12364 or anything else. Point is I could fit a different function to those points that would always produce a different 8th number. Still we try to guess at the simplest form that the questioner meant.

Hi bobbym
Cool, thanks for that. What do you think would be the next number after 30 and why? This is annoying me.

bobbym · 2010-09-12 16:36:05

Hi nha;

I like the idea of your Tribonacci sequence for the first one also.

For instance for the second sequence, I could fit a polynomial through that or I can fit this:

Each would get a different answer for the next 2. The above one thinks the next 2 are 49 and 80.
The above sequence was changed to the set to { 1,2,4,7,11,19,30...}

Now to me the sequence { 2,1,4,7,11,19,30...} looks like a typo, last 2 should be 18 , 29 instead of 19 and 30 making it a lucas type recurrence.

My guess is the next one is 52. But it is just a guess based on a weak pattern.

nha · 2010-09-12 17:03:18

bobbym wrote:

My guess is the next one is 52. But it is just a guess based on a weak pattern.

Thanks again bobbym. How did you get 52, what weak pattern did you find? I couldn't find any sort of pattern.

bobbym · 2010-09-12 17:17:53

Hi;

Sorry, a total hallucination. I am just not getting it right now. Are you sure that is the correct sequence?

nha · 2010-09-12 17:25:45

bobbym wrote:

Hi;
Sorry, a total hallucination. I am just not getting it right now. Are you sure that is the correct sequence?

That is how it is stated but I am starting to think that it might be a typo or a trick question. I can't seem to get it and neither can any of my friends.

bobbym · 2010-09-12 17:28:25

Hi nha;

It looks like a typo to me too. Maybe somebody else will see something I am missing. Sorry, for not getting it.

bobbym · 2010-09-12 18:45:27

Hi soroban;

I checked by solving it another way and see it the answers match up.
We are trying a a third degree polynomial to 4 data points a,b,c,d. We get the 4 x 4 set of simultaneous equations.

remembering we used n = 1,2,3,4 in f(n) as our sampled values.

There are differences between mine and yours. I checked yours with a sample data set:

Your answer gets this polynomial, if I am using it right...

Using a = 2, b = 5, c = 11, d = 23, I get:

The correct answer is:

Please see if you can discover where you or I have gone wrong.

soroban · 2010-09-14 00:37:42

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bobbym · 2010-09-14 01:17:40

Cool method! Here is another, I call it the nought sock. It will a fit a
sequence in terms of ncr's.

Starting with the sequence:

Form the difference table.

Use the leftmost element of each row as the coefficient.

This one predicts f(6) = 298 while yours predicts 251. This shows that the next term in such sequences is mathematically arbitrary.

soroban · 2010-09-14 04:09:08

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bobbym · 2010-09-14 08:57:15

Hi soroban;

Interpolation or curve fitting is nice. I guess everybody who loves math starts with looking at some piece of it and saying,"I wish I could do that, that is great." For me it was integration and curve fitting. What was yours?

Math Is Fun Forum

#1 2010-09-11 23:34:47

sequence question

#2 2010-09-12 04:38:14

Re: sequence question

#3 2010-09-12 15:59:49

Re: sequence question

#4 2010-09-12 16:19:14

Re: sequence question

#5 2010-09-12 16:25:55

Re: sequence question

#6 2010-09-12 16:34:01

Re: sequence question

#7 2010-09-12 16:36:05

Re: sequence question

#8 2010-09-12 17:03:18

Re: sequence question

#9 2010-09-12 17:17:53

Re: sequence question

#10 2010-09-12 17:25:45

Re: sequence question

#11 2010-09-12 17:28:25

Re: sequence question

#12 2010-09-12 18:45:27

Re: sequence question

#13 2010-09-14 00:37:42

Re: sequence question

#14 2010-09-14 01:17:40

Re: sequence question

#15 2010-09-14 04:09:08

Re: sequence question

#16 2010-09-14 08:57:15

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