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A031361
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Number of symmetrically inequivalent coincidence rotations of index n in lattice Z^4.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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Dirichlet series: (1+2^(1-s))* Product (1+p^(-s))*(1+p^(1-s))/((1-p^(1-s))*(1-p^(2-s))); p != 2.
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MAPLE
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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 ;
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MATHEMATICA
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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]]];
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CROSSREFS
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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EXTENSIONS
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Typo in formula (exclamation mark for 1) corrected by R. J. Mathar, Jul 23 2010
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STATUS
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approved
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