What happens to amortized analysis in binary counter if flipping a bit at index k costs now 2^k instead of 1?

Suppose that flipping bit #i costs 2ⁱ; so, flipping bit #0 costs 1, flipping bit #1 costs 2, flipping bit #2 costs 4, and so on.

What is the amortized cost of a call to Increment(), if we call it n times?

I think the cost for n Increments should be n·2⁰/2⁰ + n·2¹/2¹ + n·2²/2² + … +n·2ⁿ⁻¹/2ⁿ⁻¹ = n(n−1) = O(n²). So each Increment should be O(n), right? But the assignment requires me to prove that it's O(log n). What have I done wrong here?

Solution

Let we have p-bit integer and walk through all 2^p possible values.

The rightmost bit is flipped 2^p times (at every step), so overall cost is 2^p
The second bit is flipped 2^(p-1) times but with cost 2, so we have overall cost 2^p
..
The most significant bit is flipped once, but with cost 2^p, so we have overall cost 2^p

Summing costs for all bits, we have full cost p*2^p for all operations.

Per-operation (amortized) cost is p*2^p / 2^p = p

But note this is cost in bit quantity, and we must express it in N terms

N = 2^p 
 so 
p = log(N)

Finally amortized complexity per operation is O(log(N))