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A332497
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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.
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5
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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
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OFFSET
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0,4
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COMMENTS
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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 |
.---.---.
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LINKS
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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]
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FORMULA
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EXAMPLE
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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.
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PROG
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(PARI) a(n) = { my (x=0, k=1); while (n, if (n%3==1, x=2^k-1-x); n\=3; k++); x }
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CROSSREFS
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See A332412 for a similar sequence.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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