A332497 a(n) = x(w+1) where x(0) = 0 and x(k+1) = 2^(k+1)-1-x(k) (resp. x(k)) when d_k = 1 (resp. d_k <> 1) and Sum_{k=0..w} d_k*3^k is the ternary representation of n. Sequence A332498 gives corresponding y's.

0, 1, 0, 3, 2, 3, 0, 1, 0, 7, 6, 7, 4, 5, 4, 7, 6, 7, 0, 1, 0, 3, 2, 3, 0, 1, 0, 15, 14, 15, 12, 13, 12, 15, 14, 15, 8, 9, 8, 11, 10, 11, 8, 9, 8, 15, 14, 15, 12, 13, 12, 15, 14, 15, 0, 1, 0, 3, 2, 3, 0, 1, 0, 7, 6, 7, 4, 5, 4, 7, 6, 7, 0, 1, 0, 3, 2, 3, 0, 1

Offset: 0

Author

Rémy Sigrist, Feb 14 2020

Keywords

• nonn
• base

Comments

The representation of {(a(n), A332498(n))} is related to the T-square fractal (see illustration in Links section).

We can iteratively build the set {(a(n), A332498(n))} as follows:

- start with X_0 = {(0, 0)},

- for k = 0, 1, ..., X_{k+1} is obtained by adjoining to X_k:

- an horizontally mirrored copy of X_k to the right,

- and a vertically mirrored copy of X_k on the top,

- this corresponds to the following substitution:

.---.

.---. | V |

| X | --> .---.---.

.---. | X | H |

.---.---.

Examples

			For n = 42:
- the ternary representation of 42 is "1120",
- x(0) = 0,
- x(1) = x(0) = 0 (as d_0 = 0 <> 1),
- x(2) = x(1) = 0 (as d_1 = 2 <> 1),
- x(3) = 2^3-1 - x(2) = 7 (as d_2 = 1),
- x(4) = 2^4-1 - x(3) = 8 (as d_3 = 1),
- hence a(42) = 8.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..6560
• Rémy Sigrist, Representation of (a(n), A332498(n)) for n = 0..3^10-1
• Rémy Sigrist, Interactive scatterplot of a 3D analog [Provided your web browser supports the Plotly library, you should see icons on the top right corner of the page: if you choose "Orbital rotation", then you will be able to rotate the plot alongside three axes]
• Wikipedia, T-square (fractal)

Crossrefs

See A332412 for a similar sequence.

Cf. A005823, A332498 (corresponding y's).

Programs

PARI

a(n) = { my (x=0, k=1); while (n, if (n%3==1, x=2^k-1-x); n\=3; k++); x }

Formula

a(n) = 0 iff n belongs to A005823.