When watching a gaming video, I noticed an intriguing fact:
$$ 1 \times 1 + 2 \times 2 + 3 \times 4 + 4 \times 8 = 49, $$
which is a square number.
I asked myself, is this a coincidence or not? Specifically, is $g(k):=\sum_{i=0}^k(i+1)2^i$ a square only for $k=0,3$? Numerical experiments up to $k=65928$ suggests that this is the case. How can I prove this?
A solution, or a source, or an idea why this might be hard, are all welcome.
The inspiration can be found early in this video (timestamp 0:22 - 0:33
).