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

Author

Orazio G. Cherubini, Jul 07 2025

Keywords

• nonn
• base
• frac

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A385706, A386237.

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

Formula

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.