Parabolic sawtooth Population is unbounded but does not tend to infinity; its graph as a function of time is a sawtooth with a parabolic envelope. The pattern works by repeating the following operation for each n>=0. A 4-glider salvo is sent southeast toward a block A, arriving in generation 20 n^2 + 144 n + a[n mod 3], where a[0]=a[2]=131 and a[1]=91. Block A is pushed 1 unit southeast and another block, B, is created upstream from A. Then, every 108 generations, 2 gliders hit B, pulling it 3 units northwest. Eventually B gets deleted by a glider, at generation 20 n^2 + 180 n + b[n mod 3], where b[0]=193, b[1]=223, and b[2]=227. Then another 4-glider salvo is sent toward A. The population is minimal around the time B is deleted. (The minimum appears to be 1208 in generations 180 n^2 + 540 n + 210 and 180 n^2 + 660n + 450.) The population is maximal around the time B is created: there are about n/30 2-glider salvos on their way toward B around generation t = 20 n^2 + 144 n, so the population is about n/3 ~ sqrt(t/180) at this time. Dean Hickerson, dean@ucdmath.ucdavis.edu 6/26/91Xref: max sawtooth

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