Zheng found an interesting loop online here (apparently blocked on campus), like this:
int m = 0,i,j; for(i = 1; i <= n ; i *= 2) for(j = 1; j <= i; j++) m++;
He tells me that the Chinese text says the complexity is O(n). I think so, too.
1. Let's start by assuming n is a power of 2. The outer loop runs through the powers of two which are less than or equal to n. So, i= 1, 2, 4, 8, 16, 32, … , n. Stated differently: i=20,21,22,23,24,,2log2n.
2. For each outer loop iteration, the number of inner loop iterations is just the index of the outer iteration. So, the total number of inner loop iterations is the sum of those outer loop index values. So, lets add them up:
20+21+22+23+24++2log2n=
log2n
Σ
k=0
2k
3. The formula cribbed from here is
n
Σ
k=0
ark=
arn+1a

r1
Applying that,
log2n
Σ
k=0
2k=
12log2n+11

21
=22log2n1=2n1=O(n)
4. Perhaps a more satifying way to look at it is to the think of doing the sum in binary. The number
20+21+22+23+24++2log2n
is a sum of binary place values. Hence, it is a binary number made of log2n+1 ones. The value of any binary number made of k ones is 2k1 (Add one to that binary number, and you'll carry through to get a 1 followed by k zeros, so the 1 is the the 2k place.) So,
2log2n+11=22log2n1=2n1=O(n)
5. If n is not a power of two, the actual number of outer-loop iterations will be the same as the largest power of two less than n. Call that number nb. But nb cannot be less than n/2, so log2nb cannot be smaller than log2n1. No threat to O(n).

Very clever. I like it.