Find the pattern: 12, 44, 130, 342, 840

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In summary, the conversation discusses a sequence of numbers and the search for a pattern or function that can accurately describe it. The numbers are taken from a game and the problem is complicated by the possibility that there may not be a single function that fits all the terms exactly. However, after some discussion and calculations, a function is proposed that fits the sequence well.
  • #1
beamthegreat
116
7
Hi I need help finding the pattern of the following sequence:

12, 44, 130, 342, 840, 1976, 4518, 10130, 22396, 48996

Any help will be appreciated. Thanks.
 
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  • #2
beamthegreat said:
Hi I need help finding the pattern of the following sequence:

12, 44, 130, 342, 840, 1976, 4518, 10130, 22396, 48996
Have you rechecked the terms carefully and if possible can you give a reference from where you have taken the problem?
 
  • #3
Are you sure that there is a pattern? I mean solved question by someone else,is it??
 
  • #4
Yes I am very certain these sequences are correct. Here are the first 20 terms:

2
12
44
130
342
840
1976
4518
10130
22396
48996
106314
229166
491280
1048304
2227918
4718250
9961092
20971100

Here's a graph of these points plotted in excel:

pattern.png
 
  • #5
Here's another graph plotted on a logarithmic scale (base 10)

pattern_log.png
 
  • #6
beamthegreat said:
Yes I am very certain these sequences are correct. Here are the first 20 terms:

2
12
44
130
342
840
1976
4518
10130
22396
48996
106314
229166
491280
1048304
2227918
4718250
9961092
20971100

Here's a graph of these points plotted in excel:

pattern.png
You have not given 2 in the question
And from where you have taken the question?
It might be that you have made up your own numbers and then asking for a pattern?
 
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  • #7
Raghav Gupta said:
You have not given 2 in the question
And from where you have taken the question?
It might be that you have made up your own numbers and then asking for a pattern?

Look, if you're not going to help me then why do you even bother posting? The pattern clearly follows an exponential trend and I need to find a method to convert a set of numbers I have into a polynomial function.
 
  • #8
Suraj M said:
Are you sure that there is a pattern? I mean solved question by someone else,is it??

Could you clarify what you mean? If its already solved by someone else then why would I post this question?

Also, if the pattern can't be expressed by any function, can anyone tell me the best fit curve to extrapolate the data? I tried using one in excel but its really bad.
 
  • #9
beamthegreat, you absolutely need to tell us where you got these numbers. If you don't, then we can't help you.
 
  • #10
Its the population growth from one of a mod in Civilization V. If your not familiar with it, its basically a turn-based computer game. I recorded the values and want to find a function for it.
 
  • #11
As I suspected it could have been a random data from game and I bothered posting because I like finding pattern in sequences but it can become hard when you take any kind of numbers.
In this case software would be helpful, giving you approximate answers as it is difficult to do by hand.
See this
 
  • #12
Raghav Gupta said:
As I suspected it could have been a random data from game and I bothered posting because I like finding pattern in sequences but it can become hard when you take any kind of numbers.
In this case software would be helpful, giving you approximate answers as it is difficult to do by hand.
See this

Yeah, but that wolframalpha program calculate a polynomial fit. But it's clear from the data that you want exponentials in there. Basically, it could be a formula of the form
[tex](\alpha x^2 + \beta x + \gamma)e^{\delta x +\varphi} + (\mu x^2 + \nu x + \sigma)[/tex]
or something very different or more complicated. If you know the exact form of the function, it's not so difficult to find the exact fit. But you don't know which functions are involved.
 
  • #13
Using cftool in matlab, I get:

f(x) = a*exp(b*x)
a = 14.61
b = 0.7462
 
  • #14
Pythagorean said:
Using cftool in matlab, I get:

f(x) = a*exp(b*x)
a = 14.61
b = 0.7462
I may be misunderstanding this but when we are giving x=1 we are getting f(x) = 30 (approximately) which is not in sequence. So is it a wrong function for the pattern?
 
  • #15
OK. I have found it. If we count the numbers from 1 upwards, the n'th number is given by [itex](n+1)(2^{n+1}-2-n) [/itex]
 
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  • #16
Raghav Gupta said:
So is it a wrong function for the pattern?
Not necessarily. There's possibility no analytical function that coincides with all terms exactly but just functions that describe general trend of growth.
 
  • #17
Svein said:
OK. I have found it. If we count the numbers from 1 upwards, the n'th number is given by [itex](n+1)(2^{n+1}-2-n) [/itex]
Wow, that looks perfectly fine.
It was seeming like a complicated function for the pattern but you showed that it is actually not.
Now a curiosity has grown in me that how you came up with it when software like Wolfram was not able to compute a formula for the sequence. Can you tell? The copyrights are to you if you have not used a software.:biggrin:
 
Last edited:
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  • #18
Raghav Gupta said:
I hope we can prove by induction that it is applicable for whole of sequence.
What are you talking about?
 
  • #19
zoki85 said:
What are you talking about?
Sorry, I was in other state of mind.
I will edit it.
 
  • #20
Raghav Gupta said:
Now a curiosity has grown in me that how you came up with it when software like Wolfram was not able to compute a formula for the sequence. Can you tell?
Well, I started out by trying to factorize the numbers. That didn't go well, but I observed that:
  • All numbers were even
  • All numbers could be factorized
That led me to try dividing all numbers by n - which did not work. But dividing by n+1 gave whole numbers as a result. Calculating the differences between the resulting numbers gave me 1, 3, 7, ... i.e. 2n-1. From there it was just a question of cleaning up the expression.
 
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  • #21
How would you know what to divide by??
 
  • #22
Equation 17 is right. This is twice the number of minimal covers of an (n+1)-set by a collection of n subsets. A cover of a set S is a collection of nonempty subsets of S whose union is S. A cover of S is called minimal if none of its proper subsets covers S.

The next one is 44039730.
 
  • #23
Svein said:
Well, I started out by trying to factorize the numbers. That didn't go well, but I observed that:
  • All numbers were even
  • All numbers could be factorized
That led me to try dividing all numbers by n - which did not work. But dividing by n+1 gave whole numbers as a result. Calculating the differences between the resulting numbers gave me 1, 3, 7, ... i.e. 2n-1. From there it was just a question of cleaning up the expression.
Brilliant guy.
Got it.
 
  • #24
OP, where did you get this question?
In general where can you get such sequences?Anybody?
 
  • #25
Suraj M said:
OP, where did you get this question?
In general where can you get such sequences?Anybody?
Refer post 12.
 
  • #26
Vanadium 50 said:
Equation 17 is right. This is twice the number of minimal covers of an (n+1)-set by a collection of n subsets. A cover of a set S is a collection of nonempty subsets of S whose union is S. A cover of S is called minimal if none of its proper subsets covers S.

The next one is 44039730.
https://oeis.org/A003469
Encyclopedia of Integer sequences is a great help with such things :wink:
Suraj M said:
In general where can you get such sequences?Anybody?
You can also use imagination to create new integer sequences.
Here is the sequence I have just made up:

1, 6, 17, 42, 97, 214, 457, 954, 1961,...

Formula is pretty simple. Try to discover it:smile:
 
  • #27
zoki85 said:
1, 6, 17, 42, 97, 214, 457, 954, 1961,...

Formula is pretty simple. Try to discover it:smile:
I tried but was not able to find a formula for such a simple thing :oldcry: . How do we tackle such kind of problems? What initial steps should we take?
 
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  • #28
zoki85 said:
Here is the sequence I have just made up:

1, 6, 17, 42, 97, 214, 457, 954, 1961,...

I almost made it last night through difference analysis. I failed to create a closed form out of it, though (I am no good late at night). So, if anybody want to clean it up:
Assuming that the first element is number one, a formula for element n is: [itex]1+\sum_{k=2}^{n}[2(2^{k}-k)+1] [/itex]
 
  • #29
Svein said:
I almost made it last night through difference analysis. I failed to create a closed form out of it, though (I am no good late at night). So, if anybody want to clean it up:
Assuming that the first element is number one, a formula for element n is: [itex]1+\sum_{k=2}^{n}[2(2^{k}-k)+1] [/itex]

Good work! The closed form is then

[tex]2^{n+2}-n^2-6[/tex]

Could I ask, how did you get that?
 
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  • #30
Mentallic said:
Could I ask, how did you get that?
  1. I copy the numbers into Excel.
  2. I let Excel calculate the differences (a2-a1, a3-a2 etc.)
  3. I let Excel calculate the differences of the differences (d2-d1, d3-d2 etc.)
  4. Repeat until I see a pattern
  5. Rebuild from the pattern upwards.
 
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  • #31
Double Exp.JPG
It is clearly a double exponential.
 
  • #32
zoki85 said:
... There's possibility no analytical function that coincides with all terms exactly ...
Example? What is the shortest such sequence?
 
  • #33
Keith_McClary said:
Example? What is the shortest such sequence?
No sequence is determined by giving finite number of terms
 
  • #34
zoki85 said:
No sequence is determined by giving finite number of terms
I thought you were saying that there might be no function that agrees with the given finite number of terms. Of course there are many such functions.
 

1. What is the pattern in the given sequence?

The pattern in the given sequence is to multiply the previous number by 3 and then subtract 8. So, starting with 12, the next number is calculated as (12 * 3) - 8 = 36 - 8 = 28. Similarly, the next number is (28 * 3) - 8 = 84 - 8 = 76 and so on.

2. Is there a mathematical formula to find the next number in the sequence?

Yes, the mathematical formula for finding the next number in the sequence is (n * 3) - 8, where n is the previous number in the sequence.

3. Can this pattern be extended to generate more numbers in the sequence?

Yes, this pattern can be extended to generate more numbers in the sequence by following the same formula of (n * 3) - 8, where n is the previous number in the sequence.

4. What is the significance of the numbers 3 and 8 in the pattern?

The numbers 3 and 8 are used in the pattern as part of the formula (n * 3) - 8. The number 3 is used to multiply the previous number, while 8 is used to subtract from the result. These numbers were specifically chosen to create a pattern that increases at a steady rate.

5. Can this pattern be applied to other sequences?

Yes, this pattern can be applied to other sequences by changing the initial number and following the same formula of (n * 3) - 8. However, the resulting sequence may not have the same pattern as the given sequence.

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