A099545 Odd part of n, modulo 4.

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, 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, 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

Offset: 1

Author

Ralf Stephan, Oct 23 2004

Keywords

• nonn
• easy

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)

Examples

			a(100) = 1: the odd part of 100 is 100/4 = 25, and 25 mod 4 = 1.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A010873, A000265, A014707, A038189.

Cf. A099544, A099546, A099547, A099548, A099549, A099550, A099551.

Programs

Mathematica

Array[Mod[#/(2^IntegerExponent[#, 2]), 4] &, 105] (* Michael De Vlieger, Feb 24 2021 *)

PARI

a(n)=bitand(n/(2^valuation(n,2)), 3); /* Joerg Arndt, Jul 18 2012 */

Python

def A099545(n): return n>>(~n&n-1).bit_length()&3 # Chai Wah Wu, Feb 26 2025

Formula

a(n) = 2 * A038189(n) + 1.

(a(n)-1)/2 = A014707(n). - Alexandre Wajnberg, Jan 02 2006

a(n) = A010873(A000265(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Aug 29 2024