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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

REFERENCES

M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

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.

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-Fran├žois Alcover, Apr 20 2020, after R. J. Mathar *)

CROSSREFS

Cf. A031360, A331140, A331141.

Sequence in context: A294195 A175982 A136666 * A155955 A221076 A230513

Adjacent sequences:  A031358 A031359 A031360 * A031362 A031363 A031364

KEYWORD

nonn,easy,nice

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 March 1 03:32 EST 2021. Contains 341732 sequences. (Running on oeis4.)