A368121 -id:A368121 - cp's OEIS frontend

A368124 A variant of A367146 with application of the distance minimization to the first of two symmetrized versions of the strip bijection between two square lattices as described in A368121.

1, 8, 12, 24, 60, 72, 168, 216, 264, 300, 624, 1560, 1692, 1752, 2232, 4824, 9804, 12456, 13080, 17064, 35040, 57084, 92184, 92952, 123096, 244584, 332652, 639192, 651432, 855240, 1660752

Offset: 1

Hugo Pfoertner, Jan 01 2024

Apparently, a(n) == 0 (mod 4) for n > 1. For cycles, whose lengths are multiples of 8, the visited points form 8 separated islands.

Larger terms are 4293336, 4462104, 5787768, 11050488, 28333080, 38414184, 72397248.

Hugo Pfoertner, Example of one of the 8 islands in orbit with L=11050488.
Hugo Pfoertner, Illustration of the orbits corresponding to the terms a(2)-a(24) of A368125, (2024).

Cf. A307110, A367146, A367150, A368121.
A368125 is a permutation of this sequence.
A368129 is the analog for the second symmetrized version of the strip bijection.

\\ Uses definitions and functions from
\\ a367150_PARI.txt and a368121_PARI.txt
cycle(v) = {my (n=1, w=BijectionD(v, BijectionK)); while (w!=v, n++; w=BijectionD(w,BijectionK)); n};
a368124(rmax=205) = {my (L=List()); for (r2=0, rmax^2, for (x=0, sqrtint(r2), my (y2=r2-x^2,y); if (issquare(y2,&y), if(x>=y, my (c=cycle([x,y])); if (setsearch(L,c)==0, print([c,[x,y],sqrt(x^2+y^2)],", "); listput(L,c); listsort(L,1)))))); L};
a368124() \\ Terms < 1000

A368125 A variant of A367894 with application of the distance minimization to the first of two symmetrized versions of the strip bijection between two square lattices as described in A368121.

Original entry on oeis.org

1, 8, 12, 60, 24, 72, 300, 264, 216, 624, 168, 1692, 2232, 1752, 4824, 1560, 9804, 17064, 13080, 35040, 12456, 57084, 123096, 92952

Offset: 1

Hugo Pfoertner, Dec 31 2023

Hugo Pfoertner, Examples of points at minimum radius.
Hugo Pfoertner, Illustration of the orbits corresponding to the terms a(2)-a(24), (2024).

Cf. A307110, A367150, A367894, A368121.
A permutation of A368124.
A368130 is the analog for the second symmetrized version of the strip bijection.

A368122 a(n) is the x-coordinate of the n-th point in a square spiral mapped to a square grid rotated by Pi/4 using the symmetrized variant of the distance-limited strip bijection described in A368121.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Jan 06 2024

sign

Hugo Pfoertner, Table of n, a(n) for n = 0..3000
Hugo Pfoertner, Plot of mapped spiral, using Plot 2.
Index entries for sequences related to coordinates of 2D curves

A368123 gives the corresponding y-coordinates.
Cf. A367150, A368121, A368127, A368128.
Analogous pair of sequences, but without symmetrization: A367895, A367896.

\\ ax(n), ay(n) after Kevin Ryde's functions in A174344 and A274923.
\\ It is assumed that the PARI programs from A367150 and A368121 have been loaded and the functions defined there are available.
ax(n) = {my (m=sqrtint(n), k=ceil(m/2)); n -= 4*k^2; if (n<0, if (n<-m, k, -k-n), if (n

A368123 a(n) is the y-coordinate of the n-th point in a square spiral mapped to a square grid rotated by Pi/4 using the symmetrized variant of the distance-limited strip bijection described in A368121.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Jan 06 2024

sign

Hugo Pfoertner, Table of n, a(n) for n = 0..3000
Index entries for sequences related to coordinates of 2D curves

A368122 gives the corresponding x-coordinates.

Cf. A367150, A368121.

\\ Identical to a368122(n), but with
a368123(n) = BijectionD([ax(n), ay(n)],BijectionK)[2];

A368126 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between Pi/4 and Pi/2 of the pair of rotated grids created by the strip bijection of A307110.

Original entry on oeis.org

0, 5, 6, 7, 8, 2, 3, 4, 1, 13, 15, 17, 19, 14, 10, 16, 11, 18, 12, 20, 9, 26, 27, 28, 25, 21, 22, 23, 24, 38, 39, 40, 41, 42, 43, 44, 37, 30, 31, 32, 33, 34, 35, 36, 29, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 61, 46, 47, 48, 45, 50, 51, 52, 53, 54, 55, 56, 49, 71, 72, 83, 74, 75, 85, 77, 78, 87, 80, 69, 81, 102, 73

Offset: 0

Hugo Pfoertner, Jan 05 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 0..10000
Klaus Nagel, Symmetrized strip bijection, with mirrored repetition of sector 1.
Hugo Pfoertner, Distance minimization applied to symmetrized strip bijection, using sector between Pi/4 and Pi/2.
Hugo Pfoertner, PARI program.

A305575, A305576 are used to enumerate the grid points in rings of increasing radius.

Cf. A307110, A367150, A368121, A368129, A368130.

PARI
```
\\ See Pfoertner link.
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cp's OEIS Frontend

A368124 A variant of A367146 with application of the distance minimization to the first of two symmetrized versions of the strip bijection between two square lattices as described in A368121.

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A368125 A variant of A367894 with application of the distance minimization to the first of two symmetrized versions of the strip bijection between two square lattices as described in A368121.

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A368122 a(n) is the x-coordinate of the n-th point in a square spiral mapped to a square grid rotated by Pi/4 using the symmetrized variant of the distance-limited strip bijection described in A368121.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A368123 a(n) is the y-coordinate of the n-th point in a square spiral mapped to a square grid rotated by Pi/4 using the symmetrized variant of the distance-limited strip bijection described in A368121.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A368126 A variant of A367150 with application of the distance minimization to an 8-fold symmetrically expanded sector between Pi/4 and Pi/2 of the pair of rotated grids created by the strip bijection of A307110.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI