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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 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. 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 Cf. A031360, A331140, A331141, A350872. Sequence in context: A294195 A175982 A136666 * A155955 A221076 A230513 Adjacent sequences: A031358 A031359 A031360 * A031362 A031363 A031364 KEYWORD nonn,easy,nice,mult AUTHOR N. J. A. Sloane 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.)