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
Links
- 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.
Programs
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Maple
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 -
Mathematica
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 *)
Formula
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)
Extensions
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
Comments