Powers of 2 Squares

There are squares that lie between 2²ⁿ and 2²⁽ⁿ⁺¹⁾
These are two consecutive even powers of two.
The difference between two consecutive squares is always some odd number 2k+1.
Give that 2²ⁿ and 2²⁽ⁿ⁺¹⁾ are two squares:
2ⁿ-1 is the number of squares between them.
2ⁿ is the number of odd number increments
2⁽²ⁿ⁺¹⁾ + 2^(2n-2) is the "middle" square
2ⁿ + 2^(n-1) is the root of 2⁽²ⁿ⁺¹⁾ + 2^(2n-2)

Example1: n = 1, 2²ⁿ = 4, 2²⁽ⁿ⁺¹⁾ = 16
2⁽²ⁿ⁺¹⁾ + 2^(2n-2) = 9, 2ⁿ + 2^(n-1) = 3

Example2: n = 2, 2²ⁿ = 16, 2²⁽ⁿ⁺¹⁾ = 64
2⁽²ⁿ⁺¹⁾ + 2^(2n-2) = 36, 2ⁿ + 2^(n-1) = 6

2²ⁿ + 2⁽ⁿ⁺¹⁾ + 1 is another square with a root of 2ⁿ + 1

2²ⁿ - 2⁽ⁿ⁺¹⁾ + 1 is another square with a root of 2ⁿ - 1