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).

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

Offset: 1

Author

Rémy Sigrist, Jan 09 2019

Keywords

• nonn

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

Crossrefs

See A323259 for the Y-coordinate.

See A163528 for a similar sequence.

Programs

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));
}