A163547 -id:A163547 - cp's OEIS frontend

A163897 a(n) = A163531(n)-A163547(n).

0, 0, 2, 4, -2, -8, -6, 0, 0, 0, 2, 16, 16, 12, 6, 0, 0, 8, 10, 24, 28, 16, 0, 0, 0, 0, 10, 28, 24, 16, 32, 40, 48, 48, 24, 20, -2, -24, -40, -36, -40, -44, -64, -60, -56, -48, -42, -56, -42, -36, -20, -16, -8, 0, 16, 8, 14, 36, 34, 24, 28, 28, 18, 24, 16, 8, -8, 0, -10

Offset: 0

Antti Karttunen, Sep 19 2009

sign

This sequence gives the difference of squares of distance from the origin to the n-th term, in the Peano curve (A163334) and the Hilbert curve (A163357) on an N x N grid. Because the Hilbert curve is based on powers of 4 and the Peano curve on powers of 9, the graph of this sequence contains dramatic swings. [Edited to Peano vs Hilbert by Kevin Ryde, Aug 28 2020]

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

Cf. A165480 (positions of 0's).

A163357 Hilbert curve in N X N grid, starting rightwards from the top-left corner, listed by descending antidiagonals.

Original entry on oeis.org

0, 1, 3, 14, 2, 4, 15, 13, 7, 5, 16, 12, 8, 6, 58, 19, 17, 11, 9, 57, 59, 20, 18, 30, 10, 54, 56, 60, 21, 23, 29, 31, 53, 55, 61, 63, 234, 22, 24, 28, 32, 52, 50, 62, 64, 235, 233, 25, 27, 35, 33, 51, 49, 67, 65, 236, 232, 230, 26, 36, 34, 46, 48, 68, 66, 78, 239, 237, 231

Offset: 0

Antti Karttunen, Jul 29 2009

			The top left 8 X 8 corner of the array shows how this surjective self-avoiding walk begins (connect the terms in numerical order, 0-1-2-3-...):
   0  1 14 15 16 19 20 21
   3  2 13 12 17 18 23 22
   4  7  8 11 30 29 24 25
   5  6  9 10 31 28 27 26
  58 57 54 53 32 35 36 37
  59 56 55 52 33 34 39 38
  60 61 50 51 46 45 40 41
  63 62 49 48 47 44 43 42

A. Karttunen, Table of n, a(n) for n = 0..32895
Jörg Arndt, Plane-filling curves on all uniform grids, arXiv:1607.02433v1 [math.CO], July 11, 2016.
Herman Haverkort, Recursive tilings and space-filling curves
Herman Haverkort, The Sound of Space-Filling Curves, Proc. Bridges 2017, pp. 399-402.
Eric Weisstein's World of Mathematics, Hilbert curve
Wikipedia, Self-avoiding walk
Wikipedia, Space-filling curve
Index entries for sequences that are permutations of the nonnegative integers

Transpose: A163359. Inverse: A163358. One-based version: A163361. Row sums: A163365. Row 0: A163482. Column 0: A163483. Central diagonal: A062880. See also A163334 & A163336 for the Peano curve.

b[{n_, k_}, {m_}] := (A[k, n] = m-1);
MapIndexed[b, List @@ HilbertCurve[4][[1]]];
Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 07 2021 *)

a(n) = A163355(A054238(n)).

Links to further derived sequences added by Antti Karttunen, Sep 21 2009

A059252 Hilbert's Hamiltonian walk on N X N projected onto x axis: m(3).

Original entry on oeis.org

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

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 23 2001

nonn

This is the X-coordinate of the n-th term in Hilbert's Hamiltonian walk A163359 and the Y-coordinate of its transpose A163357.

			[m(1)=0 0 1 1, m'(1)= 0 1 10] [m(2) =0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, m'(2)=0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3].

Antti Karttunen, Table of n, a(n) for n = 0..65535
Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.1 "The Hilbert curve", page 85, lin2hilbert.
Michael Beeler, R. William Gosper, and Richard Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, February 1972. Item 115 by Gosper, page 52. Also HTML transcription. (To use algorithm S or the state table, pad n with high 0-bits to a multiple of 4 bits.)
J. Shallit, Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences, arXiv:2106.01062 [cs.FL], June 2 2021.
Index entries for sequences related to coordinates of 2D curves

See also the y-projection, m'(3), A059253, as well as: A163539, A163540, A163542, A059261, A059285, A163547 and A163529.

void h(unsigned int *x, unsigned int *y, unsigned int l){
x[0] = y[0] = 0; unsigned int *t = NULL; unsigned int n = 0, k = 0;
for(unsigned int i = 1; i>(2*n)){
case 1: x[i] = y[i&k]; y[i] = x[i&k]+(1<Jared Rager, Jan 09 2021 */
(C++) See Fxtbook link.

Initially [m(0) = 0, m'(0) = 0]; recursion: m(2n + 1) = m(2n).m'(2n).f(m'(2n), 2n).c(m(2n), 2n + 1); m'(2n + 1) = m'(2n).f(m(2n), 2n).f(m(2n), 2n).mir(m'(2n)); m(2n) = m(2n - 1).f(m'(2n - 1), 2n - 1).f(m'(2n - 1), 2n - 1).mir(m(2n - 1)); m'(2n) = m'(2n - 1).m(2n - 1).f(m(2n - 1), 2n - 1).c(m'(2n - 1), 2n); where f(m, n) is the alphabetic morphism i := i + 2^n [example: f(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 2) = 4 4 5 5 6 7 7 6 6 7 7 6 5 5 4 4]; c(m, n) is the complementation to 2^n - 1 alphabetic morphism [example: c(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 3) = 7 7 6 6 5 4 4 5 5 4 4 5 6 6 7 7]; and mir(m) is the mirror operator [example: mir(0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3) = 3 2 2 3 3 3 2 2 1 1 0 0 0 1 1 0].

a(n) = A002262(A163358(n)) = A025581(A163360(n)) = A059906(A163356(n)).

Extended by Antti Karttunen, Aug 01 2009

A059253 Hilbert's Hamiltonian walk on N X N projected onto y axis: m'(3).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 6, 6, 7, 7, 6, 5, 5, 4, 4, 4, 4, 5, 5, 6, 7, 7, 6, 6, 7, 7, 6, 5, 5, 4, 4, 3, 2, 2, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 3, 3, 2, 2, 3, 3, 2, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 2, 2, 3, 4, 5, 5, 4, 4, 4

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 23 2001

nonn

This is the Y-coordinate of the n-th term in the type I Hilbert's Hamiltonian walk A163359 and the X-coordinate of its transpose A163357.

Antti Karttunen, Table of n, a(n) for n = 0..65535
J. Shallit, Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences, arXiv:2106.01062 [cs.FL], June 2 2021.
Index entries for sequences related to coordinates of 2D curves

See also the y-projection, m(3), A059252 as well as A163538, A163540, A163542, A059261, A059285, A163547 and A163528.

C
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See A059252.
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Initially [m(0) = 0, m'(0) = 0]; recursion: m(2n + 1) = m(2n).m'(2n).f(m'(2n), 2n).c(m(2n), 2n + 1); m'(2n + 1) = m'(2n).f(m(2n), 2n).f(m(2n), 2n).mir(m'(2n)); m(2n) = m(2n - 1).f(m'(2n - 1), 2n - 1).f(m'(2n - 1), 2n - 1).mir(m(2n - 1)); m'(2n) = m'(2n - 1).m(2n - 1).f(m(2n - 1), 2n - 1).c(m'(2n - 1), 2n); where f(m, n) is the alphabetic morphism i := i + 2^n [example: f(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 2) = 4 4 5 5 6 7 7 6 6 7 7 6 5 5 4 4]; c(m, n) is the complementation to 2^n - 1 alphabetic morphism [example: c(0 0 1 1 2 3 3 2 2 3 3 2 1 1 0 0, 3) = 7 7 6 6 5 4 4 5 5 4 4 5 6 6 7 7]; and mir(m) is the mirror operator [example: mir(0 1 1 0 0 0 1 1 2 2 3 3 3 2 2 3) = 3 2 2 3 3 3 2 2 1 1 0 0 0 1 1 0].

a(n) = A025581(A163358(n)) = A002262(A163360(n)) = A059905(A163356(n)).

Extended by Antti Karttunen, Aug 01 2009

A163531 The square of the distance from the origin to the n-th point in the Peano curve A163334.

Original entry on oeis.org

0, 1, 4, 5, 2, 1, 4, 5, 8, 13, 20, 29, 26, 17, 10, 9, 16, 25, 36, 49, 64, 65, 50, 37, 40, 53, 68, 73, 58, 45, 52, 65, 80, 89, 74, 61, 50, 41, 34, 25, 32, 41, 34, 25, 18, 13, 10, 9, 16, 17, 20, 29, 26, 25, 36, 37, 40, 53, 50, 49, 64, 65, 68, 73, 80, 89, 74, 65, 58, 45, 52, 61

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

Antti Karttunen, Table of n, a(n) for n = 0..59048

A059261 Hilbert's Hamiltonian walk on N X N projected onto the first diagonal: M(3) (sum of the sequences A059252 and A059253).

Original entry on oeis.org

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

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 24 2001

nonn

The interest comes from a simplest recursion than the cross-recursion, dependent on parity, governing the projections onto the x and y axis.

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

Cf. the x-projection m(3), A059252 and the y-projection m'(3), A059253. See also: A163530, A059285, A163547.

Initially, M(0)=0; recursion: M(n+1)=M(n).f(M(n), n).f(M(n), n+1).d(M(n), n); -f(m, n) is the alphabetic morphism i := i+2^n; [example: f(0 1 2 1 2 3 4 3 4 5 6 5 4 3 2 3, 2)=4 5 6 5 6 7 8 7 8 9 10 9 8 7 6 7 ] -d(m, n) is the complementation to 2^(n-1)*3-2, alphabetic morphism; [example: d(0 1 2 1 2 3 4 3 4 5 6 5 4 3 2 3, 3)=10 9 8 9 8 7 6 7 6 5 4 5 6 7 8 7] Here is M(3). [M(1)=0.1.2.1, M(2)=0 1 2 1.2 3 4 3.4 5 6 5.4 3 2 3]

Extended by Antti Karttunen, Aug 01 2009

cp's OEIS Frontend

A163897 a(n) = A163531(n)-A163547(n).

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A163357 Hilbert curve in N X N grid, starting rightwards from the top-left corner, listed by descending antidiagonals.

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A059252 Hilbert's Hamiltonian walk on N X N projected onto x axis: m(3).

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C

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A059253 Hilbert's Hamiltonian walk on N X N projected onto y axis: m'(3).

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C

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A163531 The square of the distance from the origin to the n-th point in the Peano curve A163334.

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A059261 Hilbert's Hamiltonian walk on N X N projected onto the first diagonal: M(3) (sum of the sequences A059252 and A059253).

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