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A323258 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). 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

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 Peano-Kurven, Elemente der Mathematik, volume 28, number 1, 1973, pages 1-10.

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(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));

}

CROSSREFS

See A323259 for the Y-coordinate.

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

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Last modified May 7 09:03 EDT 2021. Contains 343636 sequences. (Running on oeis4.)