My e-addy changed recently, and I had some math-fun and seqfan mail bounce. I am not sure if this one got through, so here goes again. Apologies for repetition if so. ---------------------------------------------------------------------------------------------------- -- First, a note on A006165: The title line %N A006165 a(2n+1) = a(n+1)+a(n), a(2n) = 2a(n). along with a(1) = 1 is clearly a recurrence for a(n) = n. It turns out that a(2) does not satify the recurrence, and we should probably indicate that: %N A006165 a(1) = a(2) = 1; a(2n) = 2a(n); a(2n+1) = a(n) + a(n+1). ---------------------------------------------------------------------------------------------------- -- Now, for an observation: Let n >= 1. Start with a bag B containing n 1's. At each step, replace the two least elements x and y in B with the single element x+y. Repeat until B contains 2 or fewer elements. Let f(n) be the largest element remaining in B at this point. Empirically, f(n) = A006165(n), it is certainly true. I was wondering if there might not be a slick proof that f satisfies the above recurrence.