A331142 -id:A331142 - cp's OEIS frontend

A331140 Number of different coincidence site lattices of index n in lattice Z^4.

1, 1, 16, 0, 36, 16, 64, 0, 152, 36, 144, 0, 196, 64, 576, 0, 324, 152, 400, 0, 1024, 144, 576, 0, 912, 196, 1392, 0, 900, 576, 1024, 0, 2304, 324, 2304, 0, 1444, 400, 3136, 0, 1764, 1024, 1936, 0, 5472, 576, 2304, 0, 3152, 912, 5184, 0, 2916, 1392, 5184, 0

Offset: 1

N. J. A. Sloane, Jan 12 2020

Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, Phil. Mag. 88 (2008), 2025-2032; arXiv:0712.0363 [math.MG]. See end of Section 2.2.

Cf. A031361, A331139, A331141, A331142.

a(2k-1) = a(4k-2) = A331139(k), a(4k) = 0. - Andrey Zabolotskiy, Jan 29 2020

Name corrected, a(2) corrected, and terms a(18) and beyond added by Andrey Zabolotskiy, Jan 29 2020

A331139 Number of different coincidence site lattices of index 2n-1 in lattice D_4.

Original entry on oeis.org

1, 16, 36, 64, 152, 144, 196, 576, 324, 400, 1024, 576, 912, 1392, 900, 1024, 2304, 2304, 1444, 3136, 1764, 1936, 5472, 2304, 3152, 5184, 2916, 5184, 6400, 3600, 3844, 9728, 7056, 4624, 9216, 5184, 5476, 14592, 9216, 6400, 12552, 7056, 11664, 14400, 8100

Offset: 1

N. J. A. Sloane, Jan 12 2020

nonn

Differs from A031360 if n is divisible by the square of an odd prime.

Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, Phil. Mag. 88 (2008), 2025-2032; arXiv:0712.0363 [math.MG]. See end of Section 2.1.
Index entries for sequences related to D_4 lattice

Cf. A031360, A331140, A331141, A331142.

Terms a(10) and beyond added and name corrected by Andrey Zabolotskiy, Jan 29 2020

A331141 Number of symmetrically inequivalent coincidence rotations of index n in lattice A_4.

Original entry on oeis.org

1, 5, 10, 20, 30, 50, 50, 80, 90, 150, 144, 200, 170, 250, 300, 320, 290, 450, 400, 600, 500, 720, 530, 800, 750, 850, 810, 1000, 900, 1500, 1024, 1280, 1440, 1450, 1500, 1800, 1370, 2000, 1700, 2400, 1764, 2500, 1850, 2880, 2700, 2650, 2210, 3200, 2450

Offset: 1

N. J. A. Sloane, Jan 12 2020

The overall number of coincidence rotations is 120 times higher. Some symmetrically distinct rotations generate the same coincidence site lattices, hence a(n) >= A331142(n). - Andrey Zabolotskiy, Jan 29 2020

Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, Phil. Mag. 88 (2008), 2025-2032; arXiv:0712.0363 [math.MG]. See end of Section 3.
Index entries for sequences related to A_4 lattice

Cf. A031360, A031361, A331139, A331140, A331142.

Name corrected and terms a(12) and beyond added by Andrey Zabolotskiy, Jan 29 2020

A350872 Number of coincidence site lattices of index n in square lattice.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0

Offset: 1

Andrey Zabolotskiy, Jan 20 2022

A coincidence site lattice (CSL), or coincidence sublattice, is a full-rank sublattice arising as an intersection of the parent lattice with its copy rotated around the origin. It is necessarily primitive.
A primitive sublattice of the square lattice is a CSL if it is square (i. e., similar to the parent lattice) and has odd index.
In this sequence, any two CSLs differing by any isometry are counted as distinct.
a(n) is also the number of ordered pairs of coprime integers (p, q) with p >= 0 and q > 0 such that p^2 + q^2 = n^2.

			a(5) = 2 index-5 CSLs have bases (2, 1), (-1, 2) and (1, 2), (-2, 1).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Baake and Peter Zeiner, Geometric enumeration problems for lattices and embedded Z-modules, arXiv:1709.07317 [math.MG], 2017; in: Aperiodic Order, vol. 2: Crystallography and Almost Periodicity, eds. M. Baake and U. Grimm, Cambridge University Press, Cambridge (2017), pp. 73-172.
Index entries for sequences related to square lattice.
Index entries for sequences related to sublattices.

Cf. A031358 (nonzero quadrisection), A004613 (positions of nonzero terms), A024362, A154269, A338690, A271102.
Cf. enumeration of wider classes of sublattices of Z^2: A000203 (all sublattices), A350871 (all well-rounded sublattices), A002654 (all square sublattices), A001615 (all primitive sublattices), A000089 (all primitive square sublattices).
Cf. enumeration of CSLs in other lattices: A331140 (Z^4), A331139 (D_4), A331142 (A_4).

csl[1] = 1;
csl[n_] := With[{f = First@Transpose@FactorInteger@n}, If[Union@Mod[f, 4] == {1}, 2^Length@f, 0]];
Array[csl, 87]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1]%4 == 1, 2, 0));} \\ Amiram Eldar, Oct 23 2023

Multiplicative with a(p^e) = 2 if p == 1 (mod 4), otherwise 0.
a(4*n+1) = A031358(n), other terms are 0.
a(n) = 2 * A024362(n) for n > 1.
Dirichlet convolution of A000089 and A154269.
Dirichlet convolution of A338690 and A271102.
From Amiram Eldar, Oct 23 2023: (Start)
Dirichlet g.f.: Product_{primes p == 1 (mod 4)} (1 + 1/p^s)/(1 - 1/p^s).
Sum_{k=1..n} a(k) = (1/Pi) * n + O(sqrt(n)*log(n)).
(both from Baake and Zeiner, 2017) (End)

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A331140 Number of different coincidence site lattices of index n in lattice Z^4.

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A331139 Number of different coincidence site lattices of index 2n-1 in lattice D_4.

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A331141 Number of symmetrically inequivalent coincidence rotations of index n in lattice A_4.

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A350872 Number of coincidence site lattices of index n in square lattice.

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