cp's OEIS Frontend

A385706 Numerator of h(n) which is the minimum among the maxima of period n cycles of T(x) = 1 - 2 * |x-1/2|.

%I A385706 #47 Jul 22 2025 16:19:12
%S A385706 0,4,6,14,26,52,106,212,426,844,1706,1126,6826,13516,27306,54062,
%T A385706 109226,216268,436906,288326,1747626,3460300,6990506,307548,27962026
%N A385706 Numerator of h(n) which is the minimum among the maxima of period n cycles of T(x) = 1 - 2 * |x-1/2|.
%C A385706 The fixed points of T^n are always rational of the form 2k/(2^n+-1) so the number of period n cycles is finite, and h(n) has this form.
%C A385706 A truncated map can be formed T_h(x)=min(h,T(x)) and h(n) is the smallest h for which this map still has a period n cycle (falling between T_0 having only the fixed point 0, and T_1 which is all of T).
%H A385706 Keith Burns and Boris Hasselblatt, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.118.03.229">The Sharkovsky Theorem: A Natural Direct Proof</a>, The American Mathematical Monthly, Vol. 118, No. 3 (2011), pp. 229-244; <a href="https://math.arizona.edu/~dwang/BurnsHasselblattRevised-1.pdf">alternative link</a>; see section 5 h(m).
%H A385706 Orazio G. Cherubini, <a href="/A385706/a385706.txt">C++ program</a>
%F A385706 a(2^n) = 2*A162634(n) for n>1 (empirical observation).
%F A385706 a(2n+1) = A020989(n) for n>1 (empirical observation).
%e A385706 For n=3: the cycles of period 3 in T are {2/7,4/7,6/7} and {2/9,4/9,8/9} with maxima 6/7 and 8/9. The minimum between those last numbers is 6/7 so a(3)=6.
%e A385706 For n=4: the cycles of period 4 in T are {2/15,4/15,8/15,14/15}, {2/17,4/17,8/17,16/17} and {6/17,12/17,10/17,14/17} with maxima 14/15,16/17,14/17. The minimum between those last numbers is 14/17 so a(4)=14.
%Y A385706 Cf. A386237 (denominators), A385708 (binary expansion).
%Y A385706 Cf. A162634, A020989.
%K A385706 nonn,more,frac
%O A385706 1,2
%A A385706 _Orazio G. Cherubini_, Jul 07 2025