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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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REFERENCES
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M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
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LINKS
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Table of n, a(n) for n=1..56.
Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, Phil. Mag. 88 (2008), 2025-2032; arXiv: preprint, 0712.0363 [math.MG], 2007.
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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 ;
# R. J. Mathar, Jul 16 2010
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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]]];
zetap (* Jean-François Alcover, Apr 20 2020, after R. J. Mathar *)
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CROSSREFS
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Cf. A031360, A331140, A331141.
Sequence in context: A294195 A175982 A136666 * A155955 A221076 A230513
Adjacent sequences: A031358 A031359 A031360 * A031362 A031363 A031364
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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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
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STATUS
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approved
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