A162634 -id:A162634 - 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|.

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

Orazio G. Cherubini, Jul 07 2025

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).

			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.

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

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

Cf. A162634, A020989.

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

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

A162647 Numerators associated with denominators A000215(n) approximating the complementary Thue-Morse constant.

Original entry on oeis.org

2, 3, 10, 151, 38506, 2523490711, 10838310072981296746, 199931532107794273605284333428918544791, 68033174967769840440887906939858451149105560803546820641877549596291376780906

Offset: 0

Vladimir Shevelev, Jul 08 2009, Jul 14 2009

If in the sequence of numbers N for which A010060(N+2^n)=1-A010060(N) the odious (evil) terms are
replaced by 1's (0's), we obtain a 2^(n+1)-periodic binary sequence. These are the post-period
binary (base-2) digits of the complementary Thue-Morse constant 1-A014571 = 0.58754596635989240221663...,
which has a continued fraction and convergents 3/5, 7/12, 10/17, 47/80, 151/257, 801/1365,...
The a(n) are numerators of the convergents selected with denominators taken from A000215.

V. Shevelev, Equations of the form t(x+a)=t(x) and t(x+a)=1-t(x) for Thue-Morse sequence, arXiv:0907.0880

Cf. A010060, A000215, A085394, A085395, A081706, A161627, A161639, A162634.

a(n)=A000215(n)-A162634(n). For n>=1, a(n+1)=1+(2^(2^n)-1)*a(n) = 1+A051179(n)*a(n).

Edited by R. J. Mathar, Sep 23 2009

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|.

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