Is there a name for a general principle behind this pattern?

[Mynahk]

Yesterday afternoon, I was doodling and noticed the following pattern (disclaimer: I do not have much mathematics background):

For any pair of consecutive natural numbers 'm' and 'n' with n > m:

1. If m and n are odd, then:
m+n = (mn+1) / P
and equivalently:
mn = P(m+n) - 1

(where 'P' is the ordinal position of the pair... for example, taken from the set of odd natural numbers, {1,3,5,7,....,2k+1}, the pair "1,3" has P=1.... " 3,5" has P=2.... "5,7" has P=3)

2. If m and n are even, then:
m+n = (mn/P) - 2
and equivalently:
mn = P(m+n+2)

_____________________________
For odd natural numbers:
P := (n-1)/2
For even natural numbers:
P:= (n/2) -1
_____________________________

Both statements (1) and (2) can be shown to be true by simple algebraic proof.
I'm wondering if anyone recognizes this pattern, and can explain it to me, or direct me to a more general principle that explains it.

[gorcee]

If you just look at the positive odd numbers, {1,3,5,...}, then for any adjacent a, b, P = (a+b)/4. This is pretty easy to see, since if b>a, and a, b are adjacent, then b = a + 2, and P = (a+a+2)/4 = (2a+2)/4 = (a+1)/2.

Then, ab = a(a+2), so ab+1 = a(a+2)+1 = a(a+1+1) + 1 = a(a+1)+a+1 = (a+1)(a+1). Then, (a+1)(a+1)/[(a+1)/2] = 2(a+1) = 2a + 2 = a+(a+2) = a+b.

There's nothing particularly special about this, afaik, since it's just manipulation of basic algebraic properties.