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

1,4

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

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

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).

Formula

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

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) ;

Crossrefs

Cf. A331143.

Sequence in context: A039457 A243749 A331143 * A068165 A086935 A077689

Adjacent sequences: A031363 A031364 A031365 * A031367 A031368 A031369

Keyword

nonn,mult

Author

N. J. A. Sloane

Extensions

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

New name from Andrey Zabolotskiy, Feb 16 2021

Status

approved