The Link Between Advanced Sequences The Likes You've Never Seen!

[CPerry]

Ok so after the showman attention-grabbing title for this thread, I do seriously need some help identifying the link between the following sequences please. Please open up the attached excel spreadsheet to view my sequences.

The problem is, this isn't a standard geometric sequence as non have a common ratio. I don't know whether it helps but underneath I have included the difference between each term.

I thought I had begun to crack it with the following formulas...
1st =(2^(n-1))-1
2rd =(2^(n-1)-2^(n-3))-1
3rd =(2^(n-1)-2^(n-2)-2^(n-5))-1

Feel free to type these into excel and you will see they work!!!... until you get to the point where the difference in the term before is bigger rather than smaller. Plotting these points in a graph, you can see this is because each sequence (going downwards) creates curved lines that have a low point before reversing.

To clarify, the sequences go down so first sequence:
1 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047

The ultimate goal is to list each formula and find the link between them for a system I am developing so this is a 2 or 3 step problem. Any assistance would be greatly appreciated and hopefully this is a bit of a fun brain scratcher for the likes of you on this forum

Callum

Sidenote: Wouldn't let me attached an excel spreadsheet so I've taken a screenshot.

 

[Dr.Peterson]

Ok so after the showman attention-grabbing title for this thread, I do seriously need some help identifying the link between the following sequences please. Please open up the attached excel spreadsheet to view my sequences.

The problem is, this isn't a standard geometric sequence as non have a common ratio. I don't know whether it helps but underneath I have included the difference between each term.

I thought I had begun to crack it with the following formulas...
1st =(2^(n-1))-1
2rd =(2^(n-1)-2^(n-3))-1
3rd =(2^(n-1)-2^(n-2)-2^(n-5))-1

Feel free to type these into excel and you will see they work!!!... until you get to the point where the difference in the term before is bigger rather than smaller. Plotting these points in a graph, you can see this is because each sequence (going downwards) creates curved lines that have a low point before reversing.

To clarify, the sequences go down so first sequence:
1 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047

The ultimate goal is to list each formula and find the link between them for a system I am developing so this is a 2 or 3 step problem. Any assistance would be greatly appreciated and hopefully this is a bit of a fun brain scratcher for the likes of you on this forum

Callum

Sidenote: Wouldn't let me attached an excel spreadsheet so I've taken a screenshot.

To clarify, can you tell us where the sequences come from, and what the goal is? Did they come to you just as a list of numbers, or do you have some way of making them from the start? And are you looking for a formula for, say, the nth term of the kth column? What sort of "link" are you looking for?

 

[CPerry]

They are weightings for a trading system I am developing as a Maths project for 'fun', more specifically they are the denominators for weightings - the numerators will be added to the formula later.

If there are 12 variables in a market then the first of those 12 will have a denominator of 2047, second will have 1535, third will have 895 etc (2047, 1535, 895, 479, 247,125, 125, 247, 479, 895, 1535, 2047) [Bottom row of data in the table]

So when a market opens on my spreadsheet I will have a value that says: There are 5 variables in this market.

I then need my system to say ok, I need to go down to the red '5' values and the first variable will have a weighting of 15 (found in the '1st' column), the second variable will have a weighting of 11 (found in the '2nd' column) etc

I need a formula that can do this based on two inputs - 1) how many variables (Red values going down the side) and 2) which position is the variable in (values going along the top).

I think I have just looked at it for so long and confused myself, not even sure which way round the sequence should be.

There can be more than 12 variables in a market, which is why it's important (and efficient for future coding) to express it as a formula rather than have it refer to this table each time)

 

[CPerry]

Apparently my last message had to be checked by a moderator. How long does this process usually take may I ask? Will this happen on each new thread I start or is it just an initial spam check since I'm new?

 

[CPerry]

Oh, that last message went straight through? I guess I’ll type up my reply again lol

The values are from a trading system that I’m playing around with as a ‘fun’ little Maths project that I’m doing just for the challenge.

The numbers are weightings (or rather the denominator of fractions for weightings) that each variable has within a market, for example if there are three variables in the market, they would be weighted: 3, 2, 3. If there were 6 then it would be: 31, 23, 13, 13, 23, 31 (read along the columns of the previously attached image).

So ultimately there will be a cell that says how many variables there are, let’s call this value n. Then if n=4, I will have 4 cells showing the values: 7, 5, 5, 7. Do you see how this is supposed to work?

As per my last message the first equation works for identifying the first variable when n is greater than 1. The other vales break down when n is smaller though??? There may be up to 30 variables in a market so you can see the purpose of having this expressed as a formula.

 

[Dr.Peterson]

Apparently my last message had to be checked by a moderator. How long does this process usually take may I ask? Will this happen on each new thread I start or is it just an initial spam check since I'm new?

You should now be past the need for moderation, and your submissions from now on will show up immediately.

Go ahead and start answering our questions.

 

[CPerry]

Thank you Dr.Peterson.

Ok so these sequences I have come up with. They are weightings (or more specifically, denominators of weightings) for variables in a trading system that I am developing for ‘fun’. I like the challenge it brings and everything up to know I have done on my own but I think I’m over thinking this part.

The goal is to have a formula that looks at how many variables are in the market i.e. n=5 and see where it is within the system i.e. 4^th position and that should return the number 11 because as you can see from the table, the row labelled 5 in red shows the numbers: 15, 11, 6, 11, 15.

Somehow I need a formula that uses the x axis value and the y axis value to return the value where the two meet from the screenshot. Upon mapping this out on a chart in excel I get a curved line for each row so I’m assuming quadratics are used.

Does anyone have any ideas please? I’m hoping there’s a pattern between the ‘1^st’ formula, ‘2^nd’ formula etc or there is one universal thing I can use for efficient code.

Many thanks in advance,

CPerry

 

[Dr.Peterson]

Ok so these sequences I have come up with. They are weightings (or more specifically, denominators of weightings) for variables in a trading system that I am developing for ‘fun’. I like the challenge it brings and everything up to know I have done on my own but I think I’m over thinking this part.

The goal is to have a formula that looks at how many variables are in the market i.e. n=5 and see where it is within the system i.e. 4^th position and that should return the number 11 because as you can see from the table, the row labelled 5 in red shows the numbers: 15, 11, 6, 11, 15.

Somehow I need a formula that uses the x axis value and the y axis value to return the value where the two meet from the screenshot. Upon mapping this out on a chart in excel I get a curved line for each row so I’m assuming quadratics are used.

Does anyone have any ideas please? I’m hoping there’s a pattern between the ‘1^st’ formula, ‘2^nd’ formula etc or there is one universal thing I can use for efficient code.

You haven't yet said where the numbers in the table come from -- how are you "developing" this system? There are clear patterns, so I assume you built those patterns in. But there are also some deviations from the patterns, which suggest you may not have been consistent. It would be much easier to make a single formula if those deviations were eliminated and each column can be clearly derived from the previous one.

You also said your formulas broke down at some point, but I don't see what you mean.

Possibly our finance-oriented members will have more to say, knowing the context.

 

[CPerry]

Ok so to describe the way I came up with these, I’ll use just a market with 6 variables as an example.

In the attached image you can see that the ‘1’ row will have: 50%, 25%, 12.5%, 6.25%, 3.125%. I theorise that the 2^nd variable will have a 50% influence on the 1^st, the 3^rd will have 25% influence etc. I then decided to ratio these by finding 1 ‘part’ by doing 50/3.125% = 16.

This gives me values of 16, 8, 4, 2, 1. The sum of these is 31 so this gives me:

1 2 3 4 5 6
X 16/31 8/31 4/31 2/31 1/31

(The sum of these = 100% or 1/1 in total and follow the rule that each variable is half as big as the previous. From my previous post you can see that row ‘6’ and column ‘1^st’ gives the value 31 (the inclusion of the numerators will come into play later).

In the second chart in the attached image you can see that the bell shape has moved giving 1 and 3 equal weighting and a space at 2. Using the same process as above I get:

1 2 3 4 5 6
8/23 X 8/23 4/23 2/23 1/23

Now when I expand this out to all 12 positions and collate the information I get the table that I previously posted and the numbers (denominators) that I am trying to come up with a calculation for, that is where the numbers come from.

I agree a single master formula for each value would be much easier. I’m simply trying to find the link between them.

When I open a new market, say there are 6 runners, I will have this information:

N=6

1^st 2^nd 3^rd 4^th 5^th 6th
1
2
3
4
5
6

From this information alone I’m hoping to develop a formula that automatically fills out the space such as the 6 variable table from my first attached image.

When I say my formula broke down try:

2^(n-1))-1

with the ‘1^st’ column’, it works perfectly, but

2^(n-1)-2^(n-2)-2^(n-5))-1

only works for values 5-12 in the ‘3^rd column but not row 3 or 4. Please see the red cells in the second attached image.

Hope this is more clear, I’m aware it is very complex but hopefully you can see what I’m trying to do and assist with condensing the process into something more manageable!

Kind regards,

CPerry.

 

[CPerry]

I'm aware this is quite an advanced concept I'm trying to pull off and it's made me scratch my head a few times already in the past few days so I really appreciate anybody taking the time to try and identify any patterns they can see. Lord knows it's left me stumped. Sincerely thank you to anyone who's currently jotting down notes and having a good think about it!

 

[CPerry]

Still could do with some help with this problem...

 

[MarkFL]

Your second column looks to me like it can be:

\(\displaystyle \left\lceil 3\cdot2^{n-3}-1\right\rceil\)

The ceiling function is only needed so that when \(\displaystyle n=2\) we get 1 returned.