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## Main Question or Discussion Point

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.

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.