interesting thing I've noted...

[karpmage]

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.

 

[daon2]

Note that (x-1)^2+x = (x-1)^2+(x-1)+1 = (x-1)(x-1+1)+1 = x(x-1)+1. It looks like you're trying to solve: 2^n-1 = m(m-1)+1, or, 2^n = m(m-1)/2 + 1

See here: http://oeis.org/A215797

There are apparently no more such numbers other than those you have listed.

 

[karpmage]

Note that (x-1)^2+x = (x-1)^2+(x-1)+1 = (x-1)(x-1+1)+1 = x(x-1)+1. It looks like you're trying to solve: 2^n-1 = m(m-1)+1, or, 2^n = m(m-1)/2 + 1

See here: http://oeis.org/A215797

There are apparently no more such numbers other than those you have listed.

Just what I was looking for. Thanks.