A031366 Number of symmetrically inequivalent coincidence rotations of icosian ring of index n.

1, 0, 0, 25, 36, 0, 0, 0, 100, 0, 288, 0, 0, 0, 0, 440, 0, 0, 800, 900, 0, 0, 0, 0, 960, 0, 0, 0, 1800, 0, 2048, 0, 0, 0, 0, 2500, 0, 0, 0, 0, 3528, 0, 0, 7200, 3600, 0, 0, 0, 2550, 0, 0, 0, 0, 0, 10368, 0, 0, 0, 7200, 0, 7688, 0, 0, 7330, 0, 0, 0, 0, 0, 0, 10368, 0, 0, 0, 0, 20000, 0, 0, 12800, 15840, 8362, 0, 0, 0, 0, 0, 0, 0, 16200, 0, 0, 0, 0, 0, 28800, 0, 0, 0, 28800, 23899

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult

Comments

The overall number of coincidence rotations is 7200 times this value. Some symmetrically distinct rotations generate the same coincidence site modules, hence a(n) >= A331143(n). - Andrey Zabolotskiy, Feb 16 2021

Links

• M. Baake, Solution of the 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.
• Michael Baake and Peter Zeiner, Coincidences in 4 dimensions, Phil. Mag. 88 (2008), 2025-2032; arXiv:0712.0363 [math.MG]. See Section 4. Caution: there is a typo in a(19).

Crossrefs

Cf. A331143.

Programs

Maple

read("transforms") :
# expansion of 1/(1-5^(-s)) in (5.10)
L1 := [1,seq(0,i=2..200)] :
for k from 1 do
    if 5^k <= nops(L1) then
        L1 := subsop(5^k=1,L1) ;
    else
        break ;
    end if;
end do:
# multiplication with 1/(1-p^(-2s)) in (5.10)
for i from 1 do
    p := ithprime(i) ;
    if modp(p,5) = 2 or modp(p,5)=3 then
        Laux := [1,seq(0,i=2..200)] :
        for k from 1 do
            if p^(2*k) <= nops(Laux) then
                Laux := subsop(p^(2*k)=1,Laux) ;
            else
                break ;
            end if;
        end do:
        L1 := DIRICHLET(L1,Laux) ;
    end if;
    if p > nops(L1) then
        break;
    end if;
end do:
# multiplication with 1/(1-p^(-s))^2 in (5.10)
for i from 1 do
    p := ithprime(i) ;
    if modp(p,5) = 1 or modp(p,5)=4 then
        Laux := [1,seq(0,i=2..200)] :
        for k from 1 do
            if p^k <= nops(Laux) then
                Laux := subsop(p^k=k+1,Laux) ;
            else
                break ;
            end if;
        end do:
        L1 := DIRICHLET(L1,Laux) ;
    end if;
    if p > nops(L1) then
        break;
    end if;
end do:
# this is now Zeta_L(s), seems to be A035187
# print(L1) ;
# generate Zeta_L(s-1)
L1shft := [seq(op(i,L1)*i,i=1..nops(L1))] ;
# generate 1/Zeta_L(s)
L1x := add(op(i,L1)*x^(i-1),i=1..nops(L1)) :
taylor(1/L1x,x=0,nops(L1)) :
L1i := gfun[seriestolist](%) ;
# generate 1/Zeta_L(2s)
L1i2 := [1,seq(0,i=2..nops(L1))] ;
for k from 2 to nops(L1i) do
    if k^2 < nops(L1i2) then
        L1i2 := subsop(k^2=op(k,L1i),L1i2) ;
    else
        break ;
    end if;
end do:
# generate Zeta_L(s)*Zeta_L(s-1)
DIRICHLET(L1,L1shft) ;
# generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s) = Phi(s)
Phis := DIRICHLET(%,L1i2) ;
# generate Phis(s-1)
Phishif := [seq(op(i,Phis)*i,i=1..nops(Phis))] ;
DIRICHLET(Phis,Phishif) ;

Formula

See Baake (1997) for the Dirichlet g.f.

Extensions

Terms beyond a(36) from R. J. Mathar, Mar 04 2018

New name from Andrey Zabolotskiy, Feb 16 2021