A350872 -id:A350872 - 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

A350871 Number of well-rounded sublattices of index n in square lattice.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 2, 2, 0, 2, 1, 2, 1, 0, 2, 0, 0, 0, 4, 3, 2, 0, 0, 2, 2, 0, 1, 0, 2, 2, 1, 2, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 4, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 8, 2, 0, 2, 1, 4, 0, 0, 2, 0, 2, 0, 1, 2, 2, 4, 0, 2, 0, 0, 6, 1, 2, 0, 2, 4, 0, 0

Offset: 1

Andrey Zabolotskiy, Jan 20 2022

nonn

A sublattice is well-rounded if the linear span of its vectors of minimal length is the whole space.
A sublattice of the square lattice is well-rounded when it is square or centered rectangular (rhombic) with not too oblong unit cell: the angles of the rhombus should be at least Pi/3.
In this sequence, any two sublattices differing by any isometry are counted as distinct.

			a(4) = 1 well-rounded index-4 sublattice has basis (2, 0), (0, 2).
a(5) = 2 w.-r. index-5 sublattices have bases (2, 1), (-1, 2) and (1, 2), (-2, 1).
At index 12, for the first time centered rectangular sublattices occur, there are a(12) = 2 of them with the bases (3, 2), (3, -2) and (2, 3), (-2, 3).

Andrey Zabolotskiy, 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. See table "Some counts of the enumeration problems for Z^2"; beware of the typo in the 60th term.
Peter Zeiner, Coincidence Site Lattices and Coincidence Site Modules, Thesis, Universität Bielefeld, 2015.
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. enumeration of other classes of sublattices of Z^2: A000203 (all sublattices), A002654 (square sublattices), A000089 (primitive square sublattices), A350872 (coincidence sublattices), A145393 (all sublattices up to isometries of the parent lattice).

Cf. A097584.

fa[s_] := Count[Divisors[s], _?(#^2 < (s/#)^2 < 3 #^2 &)];
f0[s_] := If[OddQ[s], 0, 2 fa[s/2]];
f1[s_] := With[{e2 = IntegerExponent[s, 2]}, 2 (-1)^e2 fa[s/2^e2]];
pr[s_] := Count[Range[s], _?(Mod[#^2 + 1, s] == 0 &)]; (*A000089*)
sq[n_] := Sum[pr[n/d^2], {d, Select[Range[n], Mod[n, #^2] == 0 &]}]; (*A002654*)
a[n_] := sq[n] + Sum[pr[n/d] (f0[d] + f1[d]), {d, Divisors[n]}];
Array[a, 87]

See Zeiner (2015), Theorem 7.3.1. [Note that the formula from Baake & Zeiner (2017) contains an error.]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions