2^2-1 = 2^2-1 = 1^2+2
2^3-1 = 3^2-2 = 2^2+3
2^5-1 = 6^2-5 = 5^2+6
2^13-1 = 91^2-90 = 90^2+91
I realise that x^2-(x-1)==(x-1)^2+(x)
2,3,5 and 13 are all the powers of mersenne primes, and are Fibonacci numbers as well.
It'd be interesting to see what's the next power of 2 that satisfies this equation.