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A246054 Numerators of the first rational corners of the mean-median map limit value m(x) for x >= 1/2. 1

%I #23 Nov 14 2014 13:22:29

%S 2911001,339,56346353,90028408624696264845,120193266020529,

%T 90022305806716382445

%N Numerators of the first rational corners of the mean-median map limit value m(x) for x >= 1/2.

%C The function m(x) is the limit value of the mean-median map. Start with 0<x<1 and find a new number x_4 so that the median of {0,x,1} equals the mean of {0,x,1,x_4}. Then find a new number x_5 so that the median of {0,x,1,x_4} equals the mean of {0,x,1,x_4,x_5}, etc. It is conjectured by Shultz and Shiflett that for all real x this process ends after finitely many steps. Cellarosi and Munday find a range of x for which this conjecture is true (improving results of Chamberland and Martelli), and an explicit piecewise affine formula for m(x). It is enough to study the case of 1/2 <= x <= 2/3. The first rational corners of the map m(x) to the right of 1/2 are found by Cellarosi and Munday and these are the rationals whose numerators are listed above.

%H F. Cellarosi, S. Munday, <a href="http://arxiv.org/abs/1408.3454">On two conjectures for M&m sequences</a>, arXiv:1408.3454 [math.CO], 2014.

%H M. Chamberland, and M. Martelli, <a href="http://www.math.grin.edu/~chamberl/papers/mean_median.pdf">The mean-median map</a>, Journal of Difference Equations and Applications 13, 577--583 (2007)

%H H. Shultz, and R. Shiflett, <a href="http://www.jstor.org/stable/30044851">M&m Sequences</a>, The College Mathematics Journal 36, Number 3, 2005.

%Y Cf. A246055 (denominators).

%K nonn,frac,more

%O 1,1

%A _Francesco Cellarosi_, Aug 12 2014

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