Orazio G. Cherubini - cp's OEIS frontend

A385708 Periodic part of the binary expansion of A385706(n) / A386237(n).

0, 1100, 110, 11010010, 11010, 110100, 1101010, 1101001100101100, 110101010, 1101001100, 11010101010, 110100110010, 1101010101010, 11010011001100, 110101010101010, 11010011001011010010110011010010, 11010101010101010, 110100110011001100, 1101010101010101010, 11010011001011010010

Offset: 1

Orazio G. Cherubini, Jul 07 2025

a(n) for n odd seems to be given by 1 followed by (n-1)/2 copies of 10 .
a(n) seems to have length n for n not a power of 2. It makes sense given that A386237(n) appears to be 2^n-1 for n not a power of 2.
a(n) seems to heve length 2n for n a power of 2. It makes sense given that A386237(n) appears to be 2^n+1 for n a power of 2.

			For n=3 A385706(3)/A386237(3)=6/7=(0.110110110...)_2 so a(3)=110.
For n=8 a(8) = a(2^3) = (a(2^2)+1)|opp(a(2^2)+1) = 11010011|00101100 = 1101001100101100.

K. Burns and B. Hasselblatt, The Sharkovsky Theorem: A Natural And Direct Proof, 2008.

Cf. A385706, A386237.

Cf. A010060 (for empirical relation on the 2^n terms).

a(2n+1) = 110...10 with n copies of 10 (empirical observation).
a(4n+2) = 1101001100...1100 with n-1 copies of 1100 (empirical observation).
a(2^(n+1)) = (a(2^n)+1)|opp(a(2^n)+1) where opp switches 1 and 0 and | denotes juxtapositions (empirical observation): this suggests a relation with A010060 which is observed also in A385706.

A386237 Denominators of h(n) which is the minimum among the maxima of period n cycles of T(x) = 1 - 2 * |x-1/2|.

Original entry on oeis.org

1, 5, 7, 17, 31, 63, 127, 257, 511, 1023, 2047, 1365, 8191, 16383, 32767, 65537, 131071, 262143, 524287, 349525, 2097151, 4194303, 8388607, 372827, 33554431

Offset: 1

Orazio G. Cherubini, Jul 16 2025

The fixed points of T^n are always rational of the form 2k/(2^n+-1) so the minimum among the maxima has again this property.
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).
It appears that a(n)=A000225(n) for n not a power of 2 when a(n) does not simplify with the numerator.
It appears that a(2^n)=A000051(2^n)=A000215(n) when a(2^n) does not simplify with the denominator.

			For n=3: the three cycles of 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)=7.
For n=4: the four cycles of 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)=17.

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.
Orazio G. Cherubini, a386237

Cf. A385706 (numerators).
Cf. A385708 (periodic part of binary expansion of A385706(n)/a(n)).
Cf. A000215,A000051,A000225 (for empirical observations).

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

Original entry on oeis.org

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).

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User: Orazio G. Cherubini

A385708 Periodic part of the binary expansion of A385706(n) / A386237(n).

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A386237 Denominators 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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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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