A031362 Number of coincidence site modules of index 10n+1 with 10-fold symmetry in plane.

1, 4, 0, 4, 4, 0, 4, 4, 0, 0, 4, 0, 8, 4, 0, 4, 0, 0, 4, 4, 0, 4, 0, 0, 4, 4, 0, 4, 4, 0, 0, 4, 0, 4, 16, 0, 0, 0, 0, 0, 4, 0, 4, 4, 0, 16, 4, 0, 0, 4, 0, 0, 4, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 4, 4, 0, 4, 16, 0, 4, 4, 0, 0, 0, 0, 4, 4, 0, 16, 0, 0, 4, 4, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 8, 4, 0, 4

Offset

1,2

Comments

The Dirichlet g.f. is Sum_{n>=0} a(n+1)/(1+10n)^s. - R. J. Mathar, Jul 16 2010

References

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

Baake, M. and P. A. B. Pleasants. "The coincidence problem for crystals and quasicrystals." Aperiodic, vol. 94, pp. 25-29. 1995.

Links

Table of n, a(n) for n=1..100.

Formula

Dirichlet series: Product ((1+p^(-s))/(1-p^(-s)))^2 (p=1 mod 5).

Maple

read("transforms") : maxOrd := 1000 :
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 := [1, seq(0, n=2..maxOrd)] : for e from 1 to maxOrd by 5 do if isprime(e) then ZetaNum(e, maxOrd, 1) ; Zetap := DIRICHLET(Zetap, %) ; Zeta(e, maxOrd, 1) ; Zetap := DIRICHLET(Zetap, %) ; end if; end do:
Zetap := DIRICHLET(Zetap, Zetap) ; seq( Zetap[1+10*e], e=0..(nops(Zetap)-1)/10) ;
# R. J. Mathar, Jul 16 2010

Mathematica

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];
maxOrd = 1000;
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 = Join[{1}, Table[0, {n, 2, maxOrd}]]; For[e = 1, e <= maxOrd, e += 5, If[PrimeQ[e], ze = zetaNum[e, maxOrd, 1];
zetap = DIRICHLET[zetap, ze]; ze = zeta[e, maxOrd, 1];
zetap = DIRICHLET[zetap, ze]]];
zetap = DIRICHLET[zetap, zetap];
Table[zetap[[1 + 10 e]], {e, 0, (Length[zetap] - 1)/10}] (* Jean-François Alcover, Apr 05 2020, after R. J. Mathar *)

Prog

(PARI)
M=1200
t4=direuler(p=2, M, (1+(p%5<2)*X)) \\ p == 0 or 1 mod 5
t5=direuler(p=2, M, 1/(1+(p%5<1)*X)) \\ p == 0  mod 5
t6=dirmul(t4, t5) \\ p == 1 mod 5
t7=direuler(p=2, M, 1/(1-(p%5<2)*X))
t8=direuler(p=2, M, (1-(p%5<1)*X))
t9=dirmul(t7, t8)
t10=dirmul(t6, t9)
t10b=dirmul(t10, t10)
t11=vector(40, n, t10b[10*n+1]) \\ (and then prepend 1)
\\ N. J. A. Sloane, Nov 15 2019

Crossrefs

Sequence in context: A016680 A062524 A152856 * A198414 A309721 A110854

Adjacent sequences: A031359 A031360 A031361 * A031363 A031364 A031365

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from R. J. Mathar, Jul 16 2010

Status

approved