An odd integer series formula?

In summary, the conversation discussed a relationship between the difference of squares of successive integers and the sums of their roots. The equation (x+y) = (y2 - x2) was expanded to include any integer n, resulting in n(x+y) = (y2 - x2). By starting with x = 1 and y = 2 and increasing both by 1 at each iteration, a sequence of odd integers was produced. This was shown to be related to the difference of squares and the equation (y-x)(y+x) = n(y+x). The conversation also touched on the idea of using functions to explain the relationship.
  • #1
Dennis Plews
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A few months ago I posted a simple equation that shows an interesting nexus between the difference between the squares of successive integers and the sums of their roots, viz:

Where y = x+1 then (x + y) = (y2 - x2)

Recently I expanded this relationship as follows:

Where n is any integer and y = (x + n), then n(x+y) = (y2 - x2)

Starting with x = 1 and y = 2 and increasing the x and y values by 1 at each iteration, this seems to produce an odd integers sequence as follows:

(1 + 2) = 3 = (4 - 1)
(2 + 3) = 5 = (9 - 4)
(3 + 4) = 7 = (16 - 9)
(4 + 5) = 9 = (25 - 16)
(5 + 6) = 11 = (36 - 25)...

Using the y = (x + n) form with the x value at 1 and increasing y by (x + n) gives a similar result:

1(1 + 2) = 3 = (4 - 1)
2(1 + 3) = 8 = (9 - 1)
3(1 + 4) = 15 = (16 - 1)
4(1 + 5) = 24 = (25 - 1)
5(1 + 6) = 35 = (36 -14)...

The difference between the successive results values being a sequence of odd integers.

Using the y = (x + n) form with the x value at 2 gives a similar result:

1(2 + 3) = 5 = (9 - 4)
2(2 + 4) = 12 = (16 - 4)
3(2 + 5) = 21 = (25 - 4)
4(2 + 6) = 32 = (36 - 4)
5(2 + 7) = 45 = (49 - 4)...

The difference between the successive results values again being a sequence of odd integers.

Not being very sophisticated mathematically I looked through Wikipedia’s integer series page (https://en.wikipedia.org/wiki/Integer_sequence) and found nothing like this series. Fibonacci numbers seem similar. I am curious to learn if this relationship is already known and whether it has any relationship to other known mathematical relationships.
 
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  • #2
To be frank here, I'm not quite sure what the interesting thing about the relationships that you found is - so if I missed it, please do point it out.

Dennis Plews said:
Starting with x = 1 and y = 2 and increasing the x and y values by 1 at each iteration, this seems to produce an odd integers sequence as follows:

(1 + 2) = 3 = (4 - 1)
(2 + 3) = 5 = (9 - 4)
(3 + 4) = 7 = (16 - 9)
(4 + 5) = 9 = (25 - 16)
(5 + 6) = 11 = (36 - 25)...
Well, if you increased x and y by 1 each, the sum (x+y) increases by 2 for each step, so if u started with an odd value, you will naturally generate a sequence of odd integers.

Dennis Plews said:
The difference between the successive results values being a sequence of odd integers.
Well the difference between successive steps is
##[(y+1)^2 - x^2] - [y^2 - x^2] = 2y + 1##
which naturally forms a sequence of successive odd integers since you're increasing ##y## by 1 each step.
 
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  • #3
Difference of squares: ##y^2-x^2=(y+x)(y-x)##. Substitute ##y=x+n## to get ##y^2-x^2=(y+x)n##.
 
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  • #4
Dennis Plews said:
A few months ago I posted a simple equation that shows an interesting nexus between the difference between the squares of successive integers and the sums of their roots, viz:

Where y = x+1 then (x + y) = (y2 - x2)

Recently I expanded this relationship as follows:

Where n is any integer and y = (x + n), then n(x+y) = (y2 - x2)
...
Yes, a little over nine months ago you started such a thread.

In that thread you discussed the equation x + y = y2 - x2 .

I assume you intend the 2 to be used as an exponent here also.

The equation ##\ y = x+n \ ## is equivalent to the equation ##\ y - x=n \ ##.

Multiplying that ##\ y+x\ ## gives ##\ (y - x)(y+x) =n(y+x) \ ##.

This equation has a set of solutions in addition to those for the initial equation. These are the solutions to ##\ y=x\ ##.

Of course, ##\ (y - x)(y+x) =n(y+x) \ ## is the same as ##\ y^2 - x^2 =n(y+x) \ ##.
 
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  • #5
Let f(x) = x2 .

What you have is variations on f(x+1) - f(x), for integer values of x.

For the case with n, it's essentially a similar difference with a cubic function.
 
  • #6
I appreciate all of your comments. I have learned something from each and am encouraged to further my math skills.
 

Related to An odd integer series formula?

1. What is an odd integer series formula?

An odd integer series formula is a mathematical equation used to find the sum of a series of consecutive odd integers. It is often represented as 1 + 3 + 5 + ... + (2n+1) and can be used to find the sum of any number of odd integers.

2. How do I find the sum of an odd integer series?

To find the sum of an odd integer series, you can use the formula n^2, where n represents the number of terms in the series. For example, if you have 10 terms in the series, the sum would be 10^2 = 100. Alternatively, you can also use the formula (n/2)(2a + (n-1)d), where a is the first term in the series, d is the common difference between terms, and n is the number of terms.

3. Can an odd integer series have a negative sum?

No, an odd integer series cannot have a negative sum. This is because the sum of an odd integer series will always be a positive number, since all of the terms in the series are positive odd integers. If you are getting a negative number as the sum, it is likely that there was a mistake in the calculations.

4. How is an odd integer series formula related to triangular numbers?

An odd integer series formula is closely related to triangular numbers, which are a series of numbers that form a triangle when arranged in a pattern. This is because the sum of an odd integer series can also be represented as a triangular number, specifically the (n+1)th triangular number, where n is the number of terms in the series.

5. What is the importance of an odd integer series formula?

An odd integer series formula is important in mathematics because it is a fundamental concept that is used to teach basic algebraic principles. It also has many real-world applications, such as in calculating the total number of items in a row or the sum of alternating currents in an electrical circuit.

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