This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385708 #44 Jul 23 2025 00:04:26 %S A385708 0,1100,110,11010010,11010,110100,1101010,1101001100101100,110101010, %T A385708 1101001100,11010101010,110100110010,1101010101010,11010011001100, %U A385708 110101010101010,11010011001011010010110011010010,11010101010101010,110100110011001100,1101010101010101010,11010011001011010010 %N A385708 Periodic part of the binary expansion of A385706(n) / A386237(n). %C A385708 a(n) for n odd seems to be given by 1 followed by (n-1)/2 copies of 10 . %C A385708 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. %C A385708 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. %H A385708 K. Burns and B. Hasselblatt, <a href="https://math.arizona.edu/~dwang/BurnsHasselblattRevised-1.pdf">The Sharkovsky Theorem: A Natural And Direct Proof</a>, 2008. %F A385708 a(2n+1) = 110...10 with n copies of 10 (empirical observation). %F A385708 a(4n+2) = 1101001100...1100 with n-1 copies of 1100 (empirical observation). %F A385708 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. %e A385708 For n=3 A385706(3)/A386237(3)=6/7=(0.110110110...)_2 so a(3)=110. %e A385708 For n=8 a(8) = a(2^3) = (a(2^2)+1)|opp(a(2^2)+1) = 11010011|00101100 = 1101001100101100. %Y A385708 Cf. A385706, A386237. %Y A385708 Cf. A010060 (for empirical relation on the 2^n terms). %K A385708 nonn,base,frac %O A385708 1,2 %A A385708 _Orazio G. Cherubini_, Jul 07 2025