Name of the phenomenon: square of previous number + odd number = square of the next number

Question

I don't know maths but I've made up a method of finding next 'square' of numbers. Have a look at this:

• 1² = 0² + 1
• 2² = 1² + 3
• 3² = 2² + 5
• 4² = 3² + 7
• 5² = 4² + 9
• 6² = 5² + 11
• 7² = 6² + 13
• 8² = 7² + 15
• 9² = 8² + 17
• 10² = 9² + 19 and so forth.

When you add the square of previous number to an odd number, it gives you the square of the next number.

I cannot explain it further as I have no knowledge about maths.

I would like to know whether it has existed or not. Does this process have a name?

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Answer 1

Say $n$ is the number you are squaring on the left. I will continuously use the example of $n=5$ that you have. So on the left, we have $n^2=5^2$. On the right you have the sum of two numbers, $(n-1)^2$ and $2n-1$, e.g. $4^2$ and $9$. Then what you have on each line is $$ n^2= (n-1)^2 + (2n-1) $$ But you can see this must always be equal just by expanding the right side $$ (n-1)^2 + (2n-1) = n^2-2n+1 + 2n-1= n^2 $$

I know of no name for this exact process, and it is unlikely to have a name for this works with any way of expanding out a polynomial without making a change to the value so you can create an infinite number of such similar expressions. The typical way such a thing would earn a name is if it had an interesting relationship/interaction with something else in Mathematics. While this may be the case, again, I do not know of the existence of such a relationship.

Answer 2

The process of finding the differences between a series of numbers such as squares is referred to as (the Calculus of) Finite Differences. The usefulness of this is not limited to squares and so its application to squares has no special name.

In the case of squares, differences are calculated twice to reach the constant $2$ (i.e finding differences and differences of differences.)

In the case of cubes calculating differences has to be done $3$ times to reach the constant $6$ or $3!$ (factorial $3!=2\times3$).

In the case of numbers to the power of 4 calculating differences has to be done $4$ times to reach the constant $24$ or $4!$ (factorial $4!=2\times3\times4$).

The pattern continues on in the same way for higher powers.

Applying finite differences, amongst other things, allows you to find if there is a power law connecting a series of numbers.

Answer 3

$ n^2-(n-1)^2=2n-1$ which is an odd number.
Not every phenomenon in maths has a name. Although we can call it a recurrence relation. But precisely such relations are used in cases of sequences. So if we consider this as a sequence, yes we can call it a recurrence relation. But in general, as far my knowledge, this thing has no specific name.