%I #28 Aug 29 2024 09:16:17
%S 1,1,3,1,1,3,3,1,1,1,3,3,1,3,3,1,1,1,3,1,1,3,3,3,1,1,3,3,1,3,3,1,1,1,
%T 3,1,1,3,3,1,1,1,3,3,1,3,3,3,1,1,3,1,1,3,3,3,1,1,3,3,1,3,3,1,1,1,3,1,
%U 1,3,3,1,1,1,3,3,1,3,3,1,1,1,3,1,1,3,3,3,1,1,3,3,1,3,3,3,1,1,3,1,1,3
%N Odd part of n, modulo 4.
%C The terms of this sequence are the even-indexed terms of A112658. - _Alexandre Wajnberg_, Jan 02 2006
%C Fractal sequence: odd terms are 1, 3, 1, 3,...; the even terms are the sequence itself: a(n)=a(2n)=a(4n)=a(8n)=a(16n)=... - _Alexandre Wajnberg_, Jan 02 2006
%C From _Micah D. Tillman_, Jan 29 2021: (Start)
%C Has the same structure as the regular paper-folding (dragon curve) sequence (A014577, A014709). We can interpret a(n) as the number of 90-degree rotations to make in a single direction at the n-th "turn" in the dragon curve. After all, making three 90-degree rotations to the left (turning a total of 270 degrees) is equivalent to making one 90-degree rotation to the right, and vice versa.
%C We can likewise produce the dragon curve by interpreting A000265(n), the whole odd part of n, as the number of 90-degree rotations to make in a single direction at the n-th "turn" in the curve. (End)
%H Michael De Vlieger, <a href="/A099545/b099545.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 2 * A038189(n) + 1.
%F (a(n)-1)/2 = A014707(n). - _Alexandre Wajnberg_, Jan 02 2006
%F a(n) = A010873(A000265(n)).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - _Amiram Eldar_, Aug 29 2024
%e a(100) = 1: the odd part of 100 is 100/4 = 25, and 25 mod 4 = 1.
%t Array[Mod[#/(2^IntegerExponent[#, 2]), 4] &, 105] (* _Michael De Vlieger_, Feb 24 2021 *)
%o (PARI) a(n)=bitand(n/(2^valuation(n,2)), 3); /* _Joerg Arndt_, Jul 18 2012 */
%Y Cf. A010873, A000265, A014707, A038189.
%Y Cf. A099544, A099546, A099547, A099548, A099549, A099550, A099551.
%K nonn,easy
%O 1,3
%A _Ralf Stephan_, Oct 23 2004