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A323258
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a(n) is the X-coordinate of the n-th point of a variation on Wunderlich's serpentine type 010 101 010 curve (starting at the origin and occupying the first quadrant).
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4
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0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 4, 3, 3, 4, 5, 5, 4, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 8, 8, 7, 7, 7, 6, 6, 6, 6, 7, 8, 8, 7, 6, 6, 7, 8, 8, 8, 8, 7, 7, 7
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OFFSET
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1,3
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COMMENTS
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The first type of Wunderlich curve is a plane-filling curve. Hence for any x >= 0 and y >= 0, there is a unique n > 0 such that a(n) = x and A323259(n) = y.
This curve form is by Robert Dickau. The curve begins with a 3x3 block of 9 points in an "S" shape. This block is replicated 9 times in an "N" pattern with rotations so the block ends are unit steps apart. The new bigger block is then likewise replicated in an N pattern, and so on. Wunderlich (see section 4 figure 3) begins instead with an N shape 3x3 block, so the curve here is the same large-scale structure but opposite 3x3 blocks throughout. - Kevin Ryde, Sep 08 2020
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LINKS
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Walter Wunderlich, Über Peano-Kurven, Elemente der Mathematik, volume 28, number 1, 1973, pages 1-10.
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PROG
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(PARI) s = [0, 1, 2, 2+I, 1+I, I, 2*I, 1+2*I, 2+2*I];
w = apply(z -> imag(z) + I*real(z), s);
r = [0, 1, 0, 3, 2, 3, 0, 1, 0]
a(n) = {
my (d=if (n>1, Vecrev(digits(n-1, 9)), [0]), z=s[1+d[1]]);
for (i=2, #d,
my (c=(3^(i-1)-1)/2*(1+I));
z = 3^(i-1) * w[1+d[i]] + c + (z-c) * I^r[1+d[i]];
);
return (real(z));
}
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CROSSREFS
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See A163528 for a similar sequence.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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