A331140 -id:A331140 - cp's OEIS frontend

A031361 Number of symmetrically inequivalent coincidence rotations of index n in lattice Z^4.

1, 2, 16, 0, 36, 32, 64, 0, 168, 72, 144, 0, 196, 128, 576, 0, 324, 336, 400, 0, 1024, 288, 576, 0, 960, 392, 1584, 0, 900, 1152, 1024, 0, 2304, 648, 2304, 0, 1444, 800, 3136, 0, 1764, 2048, 1936, 0, 6048, 1152, 2304, 0, 3248, 1920, 5184, 0, 2916, 3168, 5184, 0

Offset: 1

N. J. A. Sloane

Dirichlet product of 1 + 2/2^s with Sum_{n>=1} A031360(n)/(2n-1)^s. - R. J. Mathar, Jul 16 2010

Some symmetrically distinct rotations generate the same coincidence site lattices, hence a(n) >= A331140(n). - Andrey Zabolotskiy, Jan 29 2020

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, Phil. Mag. 88 (2008), 2025-2032; arXiv: preprint, 0712.0363 [math.MG], 2007.
M. Baake, "Solution of coincidence problem in dimensions d<=4", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44. arXiv:math/0605222 [math.MG], 2006.

Cf. A031360, A331140, A331141, A350872.

read("transforms") : maxOrd := 120 :
ZetaNum := proc(p,nmax,f) local n ; L := [1,seq(0,n=2..p-1),f,seq(0,n=p+1..nmax)] ; end proc:
Zeta := proc(p,nmax,f) local L,e; L := [1,seq(0,n=2..nmax)] ; for e from 1 do if p^e > nmax then break; else L := subsop(p^e=f^e,L) ; end if; end do: L ; end proc:
Zetap := ZetaNum(2,maxOrd,2): for e from 3 to maxOrd do if isprime(e) then ZetaNum(e,maxOrd,1) ; Zetap := DIRICHLET(Zetap,%) ; ZetaNum(e,maxOrd,e) ; Zetap := DIRICHLET(Zetap,%) ; Zeta(e,maxOrd,e) ; Zetap := DIRICHLET(Zetap,%) ; Zeta(e,maxOrd,e^2) ; Zetap := DIRICHLET(Zetap,%) ; end if; end do: Zetap ;
# R. J. Mathar, Jul 16 2010

maxOrd = 120;
did[m_, n_] := If[Mod[m, n] == 0, 1, 0];
DIRICHLET[a_List, b_List] := Module[{c = {}, i, s, d}, For[i = 1, i <= Min[Length[a], Length[b]], i++, s = 0; For[d = 1, d <= i, d++, If[did[i, d] == 1, s = s + a[[d]] b[[i/d]]]]; c = Append[c, s]]; c];
zetaNum[p_, nmax_, f_] := Module[{n}, L = Join[{1}, Table[0, {n, 2, p-1}], {f}, Table[0, {n, p+1, nmax}]]];
zeta[p_, nmax_, f_] := Module[{L, e}, L = Join[{1}, Table[0, {n, 2, nmax}] ]; For[e = 1, True, e++, If[p^e > nmax, Break[], L = ReplacePart[L, p^e -> f^e]]]; L];
zetap = zetaNum[2, maxOrd, 2];
For[e = 3, e <= maxOrd, e++, If[PrimeQ[e], ze = zetaNum[e, maxOrd, 1];
  zetap = DIRICHLET[zetap, ze]; ze = zetaNum[e, maxOrd, e];
  zetap = DIRICHLET[zetap, ze]; ze = zeta[e, maxOrd, e];
  zetap = DIRICHLET[zetap, ze]; ze = zeta[e, maxOrd, e^2];
  zetap = DIRICHLET[zetap, ze]]];
zetap (* Jean-François Alcover, Apr 20 2020, after R. J. Mathar *)

Dirichlet series: (1+2^(1-s))* Product (1+p^(-s))*(1+p^(1-s))/((1-p^(1-s))*(1-p^(2-s))); p != 2.
From Vaclav Kotesovec, Jul 18 2025: (Start)
Dirichlet g.f.: (2^s-4) * (2^s-2) * zeta(s-2) * zeta(s-1)^2 * zeta(s) / (2^s * (2^s+1) * zeta(2*s) * zeta(2*s-2)).
Sum_{k=1..n} a(k) ~ 525 * zeta(3) * n^3 / (2*Pi^6). (End)

More terms from R. J. Mathar, Jul 16 2010
Typo in formula (exclamation mark for 1) corrected by R. J. Mathar, Jul 23 2010
Name corrected 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

A031358 Number of coincidence site lattices of index 4n+1 in lattice Z^2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Andrey Zabolotskiy, Table of n, a(n) for n = 0..999
M. Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44; arXiv:math/0605222 [math.MG], 2006.
Michael Baake and Peter A. B. Pleasants, Algebraic solution of the coincidence problem in two and three dimensions, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See annotated scan of page 713.

Cf. A175647, A031359, A331140, A106594, A094178 (positions of nonzero terms).

t1=direuler(p=2,1200,(1+(p%4<2)*X))
t2=direuler(p=2,1200,1/(1-(p%4<2)*X))
t3=dirmul(t1,t2)
t4=vector(200,n,t3[4*n+1]) \\ and then prepend 1

Dirichlet series: Product_{primes p == 1 mod 4} (1+p^(-s))/(1-p^(-s)).

a(n) = 2*A106594(n) for n > 0. - Andrey Zabolotskiy, Jan 30 2020

More terms from N. J. A. Sloane, Mar 13 2009
Added condition that p must be prime to the Dirichlet series. - N. J. A. Sloane, May 26 2014
Offset corrected by Andrey Zabolotskiy, Jan 30 2020

A331142 Number of different coincidence site lattices of index n in lattice A_4.

Original entry on oeis.org

1, 5, 10, 20, 6, 50, 50, 80, 90, 30, 144, 200, 170, 250, 60, 320, 290, 450, 400, 120, 500, 720, 530, 800, 150, 850, 810, 1000, 900, 300, 1024, 1280, 1440, 1450, 300, 1800, 1370, 2000, 1700, 480, 1764, 2500, 1850, 2880, 540, 2650, 2210, 3200, 2450, 750, 2900

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 3.
Manuela Heuer and Peter Zeiner, CSLs of the root lattice A_4, J. Phys.: Conf. Ser. 226 (2010), 012024; arXiv:1301.2001 [math.MG].
Index entries for sequences related to A_4 lattice

Cf. A031360, A031361, A331139, A331140, A331141.

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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A031361 Number of symmetrically inequivalent coincidence rotations 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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A031358 Number of coincidence site lattices of index 4n+1 in lattice Z^2.

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A331142 Number of different coincidence site lattices 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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