On Tue, 11 Feb 2003, Jud McCranie wrote:

At 09:43 AM 2/11/2003, you wrote:

...

However based on looking at the first 2000 terms I boldly conjecture that there are only 4 instances where u(n)-u(n-1) = 1.

I tested that up to n=5,000,000 in June 2001, and commented about it in A002858.

Oh, I missed your comment. Heuristically this seems clear: since u(1) = 1 one of the sums of u(1), ..., u(n-1) will be u(n-1) + 1. So only extreme bad luck would keep one from having one more sum = u(n-1) + 1. The trouble with this is that the same argument holds for 2 yet there is no scarcity of n such that u(n) = u(n-1) + 2. Do you still have the sequence up to n = 5 million? If so I would be interested in knowing whether or not the frequency of i such that u(i) = u(i-1) + 2 continues to hold up. Gaps between successive such i are quite small in my limited data (up to n = 2000). Also u(i) = u(i-1) + 4 only occurs in my data for i = 18 and 28. And u(i) = u(i-1) + 13 only occurs once (at i = 53) in my list. -Edwin