A367150 -id:A367150 - cp's OEIS frontend

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.

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.

A368129 A variant of A367146 with application of the distance minimization to the second of two symmetrized versions of the strip bijection between two square lattices as described in A368126.

Original entry on oeis.org

1, 8, 12, 24, 72, 156, 168, 216, 264, 624, 1560, 1752, 1836, 2232, 4824, 12456, 13080, 16380, 17064, 35040, 92184, 92952, 123096, 128844, 244584, 639192, 651432, 855240, 945756

Offset: 1

Hugo Pfoertner, Jan 03 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 1660752, 4293336, 4462104, 5787768, 6647916, 11050488, 28333080, 38414184, 45366204, 184427544.

			See files linked in A368130 for visualization of orbits.

Hugo Pfoertner, Example of one of the 8 islands in orbit with L=184427544.

Cf. A307110, A367146, A367150, A368126.
A368130 is a permutation of this sequence.
A368124 is the analog for the first symmetrized version of the strip bijection.

\\ Uses definitions and functions from
\\ a367150_PARI.txt and a368126_PARI.txt
cycle(v) = {my (n=1, w=BijectionD(v, Bijectionk)); while (w!=v, n++; w=BijectionD(w, Bijectionk)); n};
a368129(rmax=235) = {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};
a368129() \\ Terms < 1000, takes 5-10 minutes CPU time

A368130 A variant of A367894 with application of the distance minimization to the second of two symmetrized versions of the strip bijection between two square lattices as described in A368126.

Original entry on oeis.org

1, 8, 12, 24, 156, 72, 216, 1836, 624, 168, 1752, 264, 16380, 4824, 1560, 13080, 2232, 128844, 35040, 12456, 92952, 17064, 945756, 244584

Offset: 1

Hugo Pfoertner, Dec 30 2023

Hugo Pfoertner, Examples of points at minimum radius.
Hugo Pfoertner, Illustration of the orbits corresponding to the terms a(2)-a(17), (2023).
Hugo Pfoertner, Illustration of the orbit with L=128844 corresponding to a(18), (2023).
Hugo Pfoertner, Illustration of the orbit with L=35040 corresponding to a(19), (2024).
Hugo Pfoertner, Illustration of the orbit with L=12456 corresponding to a(20), (2024).
Hugo Pfoertner, Illustration of the orbit with L=92952 corresponding to a(21), (2024).
Hugo Pfoertner, Illustration of the orbit with L=17064 corresponding to a(22), (2024).
Hugo Pfoertner, Illustration of the orbit with L=945756 corresponding to a(23), showing only every 11th visited point, (2024).
Hugo Pfoertner, Illustration of the orbit with L=244584 corresponding to a(24), showing only every 11th visited point, (2024).

Cf. A307110, A367150, A367894, A368126, A368129.
A permutation of A368129.
A368125 is the analog for the first symmetrized version of the strip bijection.

A368127 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 A368126.

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

Offset: 0

Hugo Pfoertner, Jan 07 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

A368128 gives the corresponding y-coordinates.
Cf. A367150, A368126.
Analogous 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 A368126 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

A368128 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 A368126.

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

Offset: 0

Hugo Pfoertner, Jan 07 2024

sign

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

A368127 gives the corresponding x-coordinates.

Cf. A367150, A368126.

\\ Identical to a368127(n), but with
a368128(n) = BijectionD([ax(n), ay(n)], Bijectionk)[2];

A386241 Decimal expansion of sqrt(5)*sin(Pi/8).

Original entry on oeis.org

8, 5, 5, 7, 0, 6, 1, 6, 8, 6, 3, 1, 2, 8, 3, 8, 4, 7, 7, 7, 4, 8, 1, 8, 0, 7, 1, 8, 2, 4, 6, 8, 3, 7, 0, 7, 3, 0, 1, 7, 0, 4, 1, 9, 3, 5, 9, 7, 3, 3, 4, 5, 4, 8, 0, 8, 7, 2, 2, 4, 2, 2, 8, 6, 4, 8, 0, 0, 9, 5, 0, 6, 5, 9, 8, 8, 2, 5, 8, 7, 5, 5, 4, 5, 0, 0, 9

Offset: 0

Hugo Pfoertner, Jul 18 2025

Upper bound of the wobbling distance S of two rotated square lattices. See A307110 and A307731 for the special case of rotation angle Pi/4. According to Jan Fricke (1999), the angle Pi/4 is the most unfavorable case, i.e., smaller bounds can be found for all other angles.

			0.8557061686312838477748180718246837073...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Hans-Georg Carstens, Walter A. Deuber, Wolfgang Thumser, and Elke Koppenrade, Geometrical Bijections in Discrete Lattices. Combinatorics, Probability and Computing, 8(1-2), 109-129, 1999.
Jan Fricke, Symmetric Graphs have symmetric Matchings, arXiv:1607.07426 [math.GR], 25 Jul 2016, Introduction.
Klaus Nagel, A Lower Bound for Double Lattice Bijections, August 17, 2025.
Index entries for algebraic numbers, degree 4

Cf. A002163, A144981, A182168, A307110, A307731, A367150.

First[RealDigits[Sqrt[5]*Sin[Pi/8],10,100]] (* Paolo Xausa, Aug 13 2025 *)

sqrt(5)*sin(Pi/8) \\ Charles R Greathouse IV, Aug 19 2025

Equals A002163*A182168.

The minimal polynomial is 8*x^4 - 40*x^2 + 25. - Joerg Arndt, Aug 02 2025

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];

cp's OEIS Frontend

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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A368129 A variant of A367146 with application of the distance minimization to the second of two symmetrized versions of the strip bijection between two square lattices as described in A368126.

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A368130 A variant of A367894 with application of the distance minimization to the second of two symmetrized versions of the strip bijection between two square lattices as described in A368126.

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A368127 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 A368126.

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Author

Keywords

Links

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PARI

A368128 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 A368126.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A386241 Decimal expansion of sqrt(5)*sin(Pi/8).

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Mathematica

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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