A163529 -id:A163529 - cp's OEIS frontend

A163530 a(n) = A163528(n)+A163529(n).

0, 1, 2, 3, 2, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 4, 5, 6, 7, 8, 9, 8, 7, 8, 9, 10, 11, 10, 9, 10, 11, 12, 13, 12, 11, 10, 9, 8, 7, 8, 9, 8, 7, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 9, 8, 7, 8, 9, 10, 11, 12, 13, 12, 11, 10, 9, 10, 11, 12, 13, 14, 15, 14, 13, 14, 15, 16, 17, 18, 19, 18, 17

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

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

A342218 The n-th and a(n)-th points of the Peano curve (A163528, A163529) are symmetrical with respect to the line X=Y.

Original entry on oeis.org

0, 5, 6, 7, 4, 1, 2, 3, 8, 45, 50, 51, 52, 49, 46, 47, 48, 53, 54, 59, 60, 61, 58, 55, 56, 57, 62, 63, 68, 69, 70, 67, 64, 65, 66, 71, 36, 41, 42, 43, 40, 37, 38, 39, 44, 9, 14, 15, 16, 13, 10, 11, 12, 17, 18, 23, 24, 25, 22, 19, 20, 21, 26, 27, 32, 33, 34, 31

Offset: 0

Rémy Sigrist, Mar 05 2021

In other words, a(n) is the unique k such that A163528(n) = A163529(k) and A163528(k) = A163529(n).

This sequence is a self-inverse permutation of the nonnegative integers.

			The Peano curve (A163528, A163529) begins as follows:
       +-----+-----+
       |6     7     8
       |
       +-----+-----+
        5     4    |3
                   |
       +-----+-----+
        0     1     2
- so a(0) = 0,
     a(1) = 5,
     a(2) = 6,
     a(3) = 7,
     a(4) = 4,
     a(8) = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, PARI program for A342218
Index entries for sequences that are permutations of the nonnegative integers

See A342217 and A342224 for similar sequences.

Cf. A163528, A163529, A338086.

PARI
```
See Links section.
```

my(table=[0,5,6,7,4,1,2,3,8]); a(n) = fromdigits(apply(d->table[d+1], digits(n,9)), 9); \\ Kevin Ryde, Mar 07 2021

a(n) = n iff n belongs to A338086.

a(n) < 9^k for any n < 9^k.

A163334 Peano curve in an n X n grid, starting rightwards from the top left corner, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

Original entry on oeis.org

0, 1, 5, 2, 4, 6, 15, 3, 7, 47, 16, 14, 8, 46, 48, 17, 13, 9, 45, 49, 53, 18, 12, 10, 44, 50, 52, 54, 19, 23, 11, 43, 39, 51, 55, 59, 20, 22, 24, 42, 40, 38, 56, 58, 60, 141, 21, 25, 29, 41, 37, 69, 57, 61, 425, 142, 140, 26, 28, 30, 36, 70, 68, 62, 424, 426, 143, 139

Offset: 0

Antti Karttunen, Jul 29 2009

			The top left 9 X 9 corner of the array shows how this surjective self-avoiding walk begins (connect the terms in numerical order, 0-1-2-3-...):
   0  1  2 15 16 17 18 19 20
   5  4  3 14 13 12 23 22 21
   6  7  8  9 10 11 24 25 26
  47 46 45 44 43 42 29 28 27
  48 49 50 39 40 41 30 31 32
  53 52 51 38 37 36 35 34 33
  54 55 56 69 70 71 72 73 74
  59 58 57 68 67 66 77 76 75
  60 61 62 63 64 65 78 79 80

Antti Karttunen, Table of n, a(n) for n = 0..13202
E. H. Moore, On Certain Crinkly Curves, Transactions of the American Mathematical Society, volume 1, number 1, 1900, pages 72-90. (And errata.) See section 7 (and in figure 3 rotate -90 degrees for the table here).
Giuseppe Peano, Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, volume 36, number 1, 1890, pages 157-160. Also EUDML (link to GDZ).
Eric Weisstein's World of Mathematics, Hilbert curve (this curve called "Hilbert II").
Wikipedia, Self-avoiding walk
Wikipedia, Space-filling curve
Index entries for sequences that are permutations of the natural numbers

Transpose: A163336. Inverse: A163335. One-based version: A163338. Row sums: A163342. Row 0: A163480. Column 0: A163481. Central diagonal: A163343.
See also: A163528-A163529, A163531, A163534, A163536, A163897.
See A163357 and A163359 for the Hilbert curve.

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

a(n) = A163332(A163328(n)).

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

Name corrected by Kevin Ryde, Aug 22 2020

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

A163528 The X-coordinate of the n-th point in the Peano curve A163334.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 01 2009

nonn

There is a 2-state automaton that accepts exactly those pairs (n,a(n)) where n is represented in base 9 and a(n) in base 3; see accompanying file a163528.pdf - Jeffrey Shallit, Aug 10 2023

Antti Karttunen, Table of n, a(n) for n = 0..59048
Jeffrey Shallit, Automaton for A163528
Index entries for sequences related to coordinates of 2D curves

Cf. A025581, A163335, A002262, A163337, A163325, A163332.

See also: A059252, A059253, A163529, A163530, A163531, A163532.

a(n) = A025581(A163335(n)) = A002262(A163337(n)) = A163325(A163332(n)).

Name corrected by Kevin Ryde, Aug 28 2020

A163332 Self-inverse permutation of the integers for constructing the Peano curve in an N X N grid.

Original entry on oeis.org

0, 1, 2, 5, 4, 3, 6, 7, 8, 15, 16, 17, 14, 13, 12, 9, 10, 11, 18, 19, 20, 23, 22, 21, 24, 25, 26, 47, 46, 45, 48, 49, 50, 53, 52, 51, 44, 43, 42, 39, 40, 41, 38, 37, 36, 29, 28, 27, 30, 31, 32, 35, 34, 33, 54, 55, 56, 59, 58, 57, 60, 61, 62, 69, 70, 71, 68, 67, 66, 63, 64, 65

Offset: 0

Antti Karttunen, Jul 29 2009

nonn

The integers [0,(3^k)-1] are confined to range [0,(3^k)-1].
From Kevin Ryde, Sep 04 2020: (Start)
To calculate a(n), write n in ternary digits n[k],..,n[0], where n[0] is the least significant digit.  Then the ternary digits of a(n) are a[j] = k^{n[j+1]+n[j+3]+...}(n[j]) where Peano's complement operator k^{s}(d) = d if s even or 2-d if s odd.
A single complement is k(d) = 2-d and the "exponent" is repeats k(k(k(...))).  Sum s = n[j+1] + n[j+3] + ... is every second digit above j, so digit j flips 0 <-> 2 when an odd number of odd digits (1's) among these.  The complement does not change digit parity so a second transformation re-complements back to the original digits and so self-inverse a(a(n)) = n.
Peano's curve is formed by digits of a(n) put alternately to x and y coordinates, so a(n) maps between the Peano curve the ternary Z-order curve per the formulas in A163528, A163529.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..59048
Giuseppe Peano, Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, volume 36, number 1, 1890, pages 157-160. Also EUDML (link to GDZ).
Index entries for sequences that are permutations of the natural numbers

Coordinates using this transform: A163528, A163529.
A163334 & A163336 give two variants of the Peano curve in an N X N grid.
Cf. A163355 (Hilbert curve).

a[n_] := FromDigits[With[{d = Reverse@IntegerDigits[n, 3]}, Reverse@Table[
  If[EvenQ@Total@d[[j+1 ;; ;; 2]], d[[j]], 2-d[[j]]], {j, Length@d}]], 3];
Array[a, 100] (* Andrey Zabolotskiy, Apr 08 2021, after Kevin Ryde *)

a(n) = my(v=digits(n,3)); for(start=2,3, my(s=0); forstep(i=start,#v,2, s+=v[i-1]; if(s%2,v[i]=2-v[i]))); fromdigits(v,3); \\ Kevin Ryde, Sep 04 2020

a(n) = A163327(A163333(A163327(n))).

Name corrected by Kevin Ryde, Aug 27 2020

A083884 a(n) = (3^(2*n) + 1) / 2.

Original entry on oeis.org

1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401

Offset: 0

Paul Barry, May 09 2003

Number of compositions of even natural numbers into n parts <= 8. - Adi Dani, May 28 2011

a(n) for n >= 1 gives the number of line segments in the n-th iteration of the Peano curve given by plotting (A163528, A163529) or by (Siromoney 1982) when parallel line segments that are connected end-to-end are counted as a single line segment. - Jason V. Morgan, Oct 08 2021

			From _Adi Dani_, May 28 2011: (Start)
a(2)=41: there are 41 compositions of even natural numbers into 2 parts <=8:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
(4,8),(8,4),(5,7),(7,5),(6,6);
(6,8),(8,6),(7,7);
(8,8).  (End)

Siromoney, R., & Subramanian, K.G. (1982). Space-filling curves and infinite graphs. Graph-Grammars and Their Application to Computer Science.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Roberto Amato, A note on Pythagorean Triples, arXiv:1912.05925 [math.HO], 2019. See Example 2.1 p. 4.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A000364, A002438, A007853, A083885, A086645.

[(3^(2*n) + 1) / 2: n in [0..20]]; // Vincenzo Librandi, Jun 16 2011

f[n_] := (3^(2n)+1)/2; Table[f@i, {i,0,20}] (* Michael De Vlieger, Jan 28 2015 *)

a(n)=(3^(2*n)+1)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(0) = 1, a(n) = 9*a(n-1) - 4.
a(n) = Sum_{k=0..n} binomial(2*n, 2*k)*4^k.
a(n) = A002438(n) / A000364(n); A000364(n) : Euler numbers.
G.f.: (1-5*x)/((1-x)*(1-9*x)).
a(n) = (3^n + 1^n + (-1)^n + (-3)^n)/4.
E.g.f.: exp(3*x) + exp(x) + exp(-x) + exp(-3*x).
Each term expresses a Pythagorean relationship, along with (a(n)-1) and a power of 3, n>0, such that sqrt((a(n))^2 - (a(n)-1)^2) = 3^n. E.g., 365^2 - 364^2 - 3^3 = 27 (the Pythagorean triangle (365, 364, 27)). - Gary W. Adamson, Jun 25 2006
a(n) = 10*a(n-1) - 9*a(n-2). - Wesley Ivan Hurt, Apr 21 2021

Additional comments from Philippe Deléham, Jul 10 2005

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

A191681 a(n) = (9^n - 1)/2.

Original entry on oeis.org

0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044

Offset: 0

Adi Dani, Jun 11 2011

Number of compositions of odd numbers into n parts < 9.
These are also the junctions of the Collatz trajectories of 2^(2k-1)-1 and 2^2k-1. - David Rabahy, Nov 01 2017
a(n) gives the number of turns in the n-th iteration of the Peano curve given by plotting (A163528, A163529) or by (Siromoney 1982). - Jason V. Morgan, Oct 08 2021

			a(2)=40: there are 40 compositions of odd numbers into 2 parts < 9:
1:  (0,1),(1,0);
3:  (0,3),(3,0),(1,2),(2,1);
5:  (0,5),(5,0),(1,4),(4,1),(2,3),(3,2);
7:  (0,7),(7,0),(1,6),(6,1),(2,5),(5,2),(3,4),(4,3);
9:  (1,8),(8,1),(2,7),(7,2),(3,6),(6,3),(4,5),(5,4);
11: (3,8),(8,3),(4,7),(7,4),(5,6),(6,5);
13: (5,8),(8,5),(6,7),(7,6);
15: (7,8),(8,7).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Adi Dani, Restricted compositions of natural numbers
Rani Siromoney and K. G. Subramanian, Space-filling curves and infinite graphs, International Workshop on Graph Grammars and Their Application to Computer Science, 1982.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A002452, A096053, A125857, A138894, A198964.

[(9^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 16 2011

Table[(9^n - 1)/2, {n, 0, 19}]
LinearRecurrence[{10,-9},{0,4},30] (* Harvey P. Dale, Jun 19 2011 *)

a(n)=9^n\2 \\ Charles R Greathouse IV, Oct 16 2015

a(0)=0, a(1)=4, a(n) = 10*a(n-1) - 9*a(n-2). - Harvey P. Dale, Jun 19 2011
G.f.: 4*x / ((x-1)*(9*x-1)). - Colin Barker, May 16 2013
a(n) = 2 * A125857(n+1) = 4 * A002452(n). - Bernard Schott, Oct 29 2021

Example corrected by L. Edson Jeffery, Feb 13 2015

A163533 The change in Y-coordinate when moving from the n-1:th to the n-th point in the Peano curve A163334.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0

Offset: 0

Antti Karttunen, Aug 01 2009

sign

Antti Karttunen, Table of n, a(n) for n = 0..59049

These are the first differences of A163529. See also: A163532, A163534, A163536.

a(0)=0, a(n) = A163529(n) - A163529(n-1).

Name corrected by Kevin Ryde, Aug 29 2020

cp's OEIS Frontend

A163530 a(n) = A163528(n)+A163529(n).

Views

Author

Keywords

Links

Crossrefs

A342218 The n-th and a(n)-th points of the Peano curve (A163528, A163529) are symmetrical with respect to the line X=Y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A163334 Peano curve in an n X n grid, starting rightwards from the top left corner, listed antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ... .

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Formula

Extensions

A163528 The X-coordinate of the n-th point in the Peano curve A163334.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A163332 Self-inverse permutation of the integers for constructing the Peano curve in an N X N grid.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A083884 a(n) = (3^(2*n) + 1) / 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views