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.
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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).
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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.
