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.
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from R. J. Mathar, Jul 16 2010
STATUS
approved