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%I #28 Jan 25 2022 09:28:13
%S 1,2,16,0,36,32,64,0,168,72,144,0,196,128,576,0,324,336,400,0,1024,
%T 288,576,0,960,392,1584,0,900,1152,1024,0,2304,648,2304,0,1444,800,
%U 3136,0,1764,2048,1936,0,6048,1152,2304,0,3248,1920,5184,0,2916,3168,5184,0
%N Number of symmetrically inequivalent coincidence rotations of index n in lattice Z^4.
%C Dirichlet product of 1 + 2/2^s with Sum_{n>=1} A031360(n)/(2n-1)^s. - _R. J. Mathar_, Jul 16 2010
%C Some symmetrically distinct rotations generate the same coincidence site lattices, hence a(n) >= A331140(n). - _Andrey Zabolotskiy_, Jan 29 2020
%H Michael Baake and Peter Zeiner, <a href="https://doi.org/10.1080/14786430701846206">Coincidences in 4 dimensions</a>, Phil. Mag. 88 (2008), 2025-2032; arXiv: <a href="https://arxiv.org/abs/0712.0363">preprint</a>, 0712.0363 [math.MG], 2007.
%H 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. <a href="https://arxiv.org/abs/math/0605222">arXiv:math/0605222</a> [math.MG], 2006.
%F Dirichlet series: (1+2^(1-s))* Product (1+p^(-s))*(1+p^(1-s))/((1-p^(1-s))*(1-p^(2-s))); p != 2.
%p read("transforms") : maxOrd := 120 :
%p ZetaNum := proc(p,nmax,f) local n ; L := [1,seq(0,n=2..p-1),f,seq(0,n=p+1..nmax)] ; end proc:
%p 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:
%p 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 ;
%p # _R. J. Mathar_, Jul 16 2010
%t maxOrd = 120;
%t did[m_, n_] := If[Mod[m, n] == 0, 1, 0];
%t 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];
%t zetaNum[p_, nmax_, f_] := Module[{n}, L = Join[{1}, Table[0, {n, 2, p-1}], {f}, Table[0, {n, p+1, nmax}]]];
%t 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];
%t zetap = zetaNum[2, maxOrd, 2];
%t For[e = 3, e <= maxOrd, e++, If[PrimeQ[e], ze = zetaNum[e, maxOrd, 1];
%t zetap = DIRICHLET[zetap, ze]; ze = zetaNum[e, maxOrd, e];
%t zetap = DIRICHLET[zetap, ze]; ze = zeta[e, maxOrd, e];
%t zetap = DIRICHLET[zetap, ze]; ze = zeta[e, maxOrd, e^2];
%t zetap = DIRICHLET[zetap, ze]]];
%t zetap (* _Jean-François Alcover_, Apr 20 2020, after _R. J. Mathar_ *)
%Y Cf. A031360, A331140, A331141, A350872.
%K nonn,easy,nice,mult
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _R. J. Mathar_, Jul 16 2010
%E Typo in formula (exclamation mark for 1) corrected by _R. J. Mathar_, Jul 23 2010
%E Name corrected by _Andrey Zabolotskiy_, Jan 29 2020
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