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A031361 Number of symmetrically inequivalent coincidence rotations of index n in lattice Z^4. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
LINKS
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.
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.
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 *)
CROSSREFS
Sequence in context: A294195 A175982 A136666 * A155955 A221076 A230513
KEYWORD
nonn,easy,nice,mult
AUTHOR
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
STATUS
approved

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Last modified June 21 06:38 EDT 2024. Contains 373540 sequences. (Running on oeis4.)