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

0, 4, 6, 14, 26, 52, 106, 212, 426, 844, 1706, 1126, 6826, 13516, 27306, 54062, 109226, 216268, 436906, 288326, 1747626, 3460300, 6990506, 307548, 27962026

Offset: 1

Author

Orazio G. Cherubini, Jul 07 2025

Keywords

• nonn
• more
• frac

Comments

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.

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).

Examples

			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.
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.
		

Links

• Keith Burns and Boris Hasselblatt, The Sharkovsky Theorem: A Natural Direct Proof, The American Mathematical Monthly, Vol. 118, No. 3 (2011), pp. 229-244; alternative link; see section 5 h(m).
• Orazio G. Cherubini, C++ program

Crossrefs

Cf. A386237 (denominators), A385708 (binary expansion).

Cf. A162634, A020989.

Formula

a(2^n) = 2*A162634(n) for n>1 (empirical observation).

a(2n+1) = A020989(n) for n>1 (empirical observation).