

A323258


a(n) is the Xcoordinate of the nth point of a variation on Wunderlich's serpentine type 010 101 010 curve (starting at the origin and occupying the first quadrant).


4



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

1,3


COMMENTS

The first type of Wunderlich curve is a planefilling 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 largescale structure but opposite 3x3 blocks throughout.  Kevin Ryde, Sep 08 2020


LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..6561
Robert Dickau, Wunderlich Curves
Rémy Sigrist, Illustration of initial terms
Wolfram Demonstrations Project, Wunderlich Curves
Walter Wunderlich, Über PeanoKurven, Elemente der Mathematik, volume 28, number 1, 1973, pages 110.
Index entries for sequences related to coordinates of 2D curves


PROG

(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(n1, 9)), [0]), z=s[1+d[1]]);
for (i=2, #d,
my (c=(3^(i1)1)/2*(1+I));
z = 3^(i1) * w[1+d[i]] + c + (zc) * I^r[1+d[i]];
);
return (real(z));
}


CROSSREFS

See A323259 for the Ycoordinate.
See A163528 for a similar sequence.
Sequence in context: A014604 A015199 A234044 * A219489 A051168 A281459
Adjacent sequences: A323255 A323256 A323257 * A323259 A323260 A323261


KEYWORD

nonn


AUTHOR

Rémy Sigrist, Jan 09 2019


STATUS

approved



