--NJA Sloane tells me "Dan Asimov's misunderstanding of Propp's question, which of course is https://oeis.org/A165911, ... Does it go into a cycle? WDS gave an inconclusive analysis. The graph of the first 1000 terms (go to the OEIS A165911 entry and click "graph") was also inconclusive. So in the night I added more terms to the b-file, and now the graph definitely is heading off to infinity. Roughly like e^(n/10)." --And I also think my analysis can be made more definitive (although still nonrigorous) as follows. problem: can Propp/Asimov's sequences f[n] = CorePart( f[n-1]+f[n-2] ) ever (with suitable starting integers) diverge to infinity without instead falling into a finite length cycle? Here CorePart(X) means the product of all the primes inside X's factorization, e.g. CorePart( 2*3*3*5*7*7*7*7 ) = 2*3*5*7. (Is that what Asimov meant?) If we model the X as "random" and ask about factors of P inside them (P=prime) then the chance X has exactly K factors of P inside it (K>=0) is exactly (P-1) * P^(-K-1) and then max(K-1, 0) of those factors need to be removed from X. So the expected additive reduction in ln(X) when we remove those factors is DELTA = DOUBLESUM( (K-1)*ln(P)*(P-1)*P^(-K-1), summed over primes P and integers K>=2 ). We can evaluate the sum over K in closed form, then the remaining sum over prime P is done numerically: DELTA = SUM( ln(P)/ [(P-1)*P], primes P ) =0.7554 meanwhile the increase in ln(X) due to the Fibonacci recurrence should be upper bounded by ln(2) = 0.6931 so it seems that the expected decrease in ln(X) ought to outweigh the expected increase due to summing, and therefore with "probability=1" this analysis would predict NON-growth to infinity and hence ultimate cycling. Exactly the opposite of Sloane's empirical computer finding. So evidently something is wrong with my cheesy probability model. And sure enough, examining Sloane's computer output https://oeis.org/A165911/b165911.txt we see that his numbers are NOT random uniform mod 2. Instead they are odd,odd,even,odd,odd,even,odd,odd,even... hence even only 1/3 of the time, not 1/2 as would naively be expected. Can other modular patterns be found (e.g. mod 3, etc)? Probably just a few such patterns will suffice to make the probability argument then reach the correct prediction. Maybe. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)