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.
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
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jan 09 2019
STATUS
approved