A163528 -id:A163528 - cp's OEIS frontend

A163480 Row 0 of A163334 (column 0 of A163336).

0, 1, 2, 15, 16, 17, 18, 19, 20, 141, 142, 143, 144, 145, 146, 159, 160, 161, 162, 163, 164, 177, 178, 179, 180, 181, 182, 1275, 1276, 1277, 1278, 1279, 1280, 1293, 1294, 1295, 1296, 1297, 1298, 1311, 1312, 1313, 1314, 1315, 1316, 1437, 1438, 1439

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

Cf. A163481 (Y axis), A037314 (Z-order X axis).

Coordinates: A163528, A163529.

a(n) = my(v=digits(n,3),s=Mod(0,2)); for(i=1,#v, if(s,v[i]+=6); s+=v[i]); fromdigits(v,9); \\ Kevin Ryde, Sep 29 2020

a(n) = A163332(A037314(n)). - Kevin Ryde, Sep 29 2020

A165466 Squared distance between n's location in A163334 array and A163359 array.

Original entry on oeis.org

0, 2, 2, 2, 2, 10, 10, 2, 0, 0, 2, 10, 20, 10, 10, 18, 32, 32, 50, 74, 100, 100, 72, 50, 32, 50, 50, 34, 20, 20, 16, 16, 16, 10, 4, 4, 2, 4, 8, 8, 8, 10, 20, 18, 20, 20, 26, 50, 50, 40, 20, 20, 20, 20, 32, 32, 34, 40, 58, 74, 100, 74, 74, 80, 80, 80, 52, 52, 50, 34, 34, 32

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Equivalently, squared distance between n's location in A163336 array and A163357 array. See example at A166043.

A. Karttunen, Table of n, a(n) for n = 0..65535

Positions of zeros: A165467. See also A166043, A165464, A163897, A163900.

a(n) = A000290(abs(A163529(n)-A059253(n))) + A000290(abs(A163528(n)-A059252(n))).

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).

Original entry on oeis.org

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

Rémy Sigrist, Jan 09 2019

nonn

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

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

See A323259 for the Y-coordinate.

See A163528 for a similar sequence.

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));
}

A332246 a(n) is the X-coordinate of the n-th point of the Minkowski sausage (or Minkowski curve). Sequence A332247 gives Y-coordinates.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 3, 3, 4, 5, 5, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 8, 7, 7, 8, 8, 8, 9, 9, 8, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12, 11, 11, 12, 13, 13, 12, 12, 13, 13, 14, 14, 14, 15, 15, 16, 16, 15, 15, 16, 17, 17, 16, 16, 15, 15, 14

Offset: 0

Rémy Sigrist, Feb 08 2020

This sequence is the real part of {f(n)} defined as:
- f(0) = 0,
- f(n+1) = f(n) + i^t(n)
  where t(n) is the number of 1's and 6's minus the number of 3's and 4's
             in the base 8 representation of n
    and i denotes the imaginary unit.
We can also build the curve by successively applying the following substitution to an initial vector (1, 0):
                        .--->.
                        ^    |
                        |    v
                   .--->.    .    .--->.
                             |    ^
                             v    |
                             .--->.

Rémy Sigrist, Table of n, a(n) for n = 0..4096
Robert Ferréol (MathCurve), Saucisse de Minkowski [in French]
Rémy Sigrist, Representation of the first 1+8^4 points of the Minkowski sausage
Wikipedia, Minkowski sausage
Index entries for sequences related to coordinates of 2D curves

See A163528, A323258 and A332204 for similar sequences.

Cf. A332247 (Y-coordinates).

{ dd = [0,1,0,-1,-1,0,1,0]; z=0; for (n=0, 75, print1 (real(z)", "); z += I^vecsum(apply(d -> dd[1+d], digits(n, #dd)))) }

a(8^k-m) + a(m) = 4^k for any k >= 0 and m = 0..8^k.

A165464 Squared distance between n's location in A163334 array and A163357 array.

Original entry on oeis.org

0, 0, 2, 4, 2, 4, 2, 0, 0, 2, 2, 4, 4, 4, 2, 0, 0, 2, 2, 4, 4, 2, 0, 0, 0, 0, 2, 4, 4, 2, 8, 10, 16, 16, 4, 2, 2, 10, 16, 10, 8, 8, 20, 20, 20, 18, 18, 32, 18, 10, 4, 2, 4, 10, 8, 2, 2, 10, 10, 4, 4, 4, 2, 10, 16, 26, 20, 10, 2, 4, 10, 18, 32, 32, 50, 52, 52, 34, 40, 58, 80, 80, 106, 146

Offset: 0

Antti Karttunen, Oct 06 2009

nonn

Equivalently, squared distance between n's location in A163336 array and A163359 array. See example at A166041.

A. Karttunen, Table of n, a(n) for n = 0..65535

Positions of zeros: A165465. See also A165466, A163897, A163900.

a(n) = A000290(abs(A163529(n)-A059252(n))) + A000290(abs(A163528(n)-A059253(n))).

cp's OEIS Frontend

A163480 Row 0 of A163334 (column 0 of A163336).

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A165466 Squared distance between n's location in A163334 array and A163359 array.

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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).

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Author

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Programs

PARI

A332246 a(n) is the X-coordinate of the n-th point of the Minkowski sausage (or Minkowski curve). Sequence A332247 gives Y-coordinates.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A165464 Squared distance between n's location in A163334 array and A163357 array.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula