OFFSET
1,3
COMMENTS
The terms of this sequence are the even-indexed terms of A112658. - Alexandre Wajnberg, Jan 02 2006
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
From Micah D. Tillman, Jan 29 2021: (Start)
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.
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)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = 2 * A038189(n) + 1.
(a(n)-1)/2 = A014707(n). - Alexandre Wajnberg, Jan 02 2006
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Aug 29 2024
EXAMPLE
a(100) = 1: the odd part of 100 is 100/4 = 25, and 25 mod 4 = 1.
MATHEMATICA
Array[Mod[#/(2^IntegerExponent[#, 2]), 4] &, 105] (* Michael De Vlieger, Feb 24 2021 *)
PROG
(PARI) a(n)=bitand(n/(2^valuation(n, 2)), 3); /* Joerg Arndt, Jul 18 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, Oct 23 2004
STATUS
approved