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%I #30 Feb 16 2021 13:13:15
%S 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,
%T 0,1800,0,2048,0,0,0,0,2500,0,0,0,0,3528,0,0,7200,3600,0,0,0,2550,0,0,
%U 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
%N Number of symmetrically inequivalent coincidence rotations of icosian ring of index n.
%C 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
%H M. Baake, <a href="https://arxiv.org/abs/math/0605222">Solution of the coincidence problem in dimensions d <= 4</a>, in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.; arXiv:math/0605222 [math.MG], 2006.
%H Michael Baake and Peter Zeiner, <a href="https://doi.org/10.1080/14786430701846206">Coincidences in 4 dimensions</a>, Phil. Mag. 88 (2008), 2025-2032; arXiv:<a href="https://arxiv.org/abs/0712.0363">0712.0363</a> [math.MG]. See Section 4. Caution: there is a typo in a(19).
%F See Baake (1997) for the Dirichlet g.f.
%p read("transforms") :
%p # expansion of 1/(1-5^(-s)) in (5.10)
%p L1 := [1,seq(0,i=2..200)] :
%p for k from 1 do
%p if 5^k <= nops(L1) then
%p L1 := subsop(5^k=1,L1) ;
%p else
%p break ;
%p end if;
%p end do:
%p # multiplication with 1/(1-p^(-2s)) in (5.10)
%p for i from 1 do
%p p := ithprime(i) ;
%p if modp(p,5) = 2 or modp(p,5)=3 then
%p Laux := [1,seq(0,i=2..200)] :
%p for k from 1 do
%p if p^(2*k) <= nops(Laux) then
%p Laux := subsop(p^(2*k)=1,Laux) ;
%p else
%p break ;
%p end if;
%p end do:
%p L1 := DIRICHLET(L1,Laux) ;
%p end if;
%p if p > nops(L1) then
%p break;
%p end if;
%p end do:
%p # multiplication with 1/(1-p^(-s))^2 in (5.10)
%p for i from 1 do
%p p := ithprime(i) ;
%p if modp(p,5) = 1 or modp(p,5)=4 then
%p Laux := [1,seq(0,i=2..200)] :
%p for k from 1 do
%p if p^k <= nops(Laux) then
%p Laux := subsop(p^k=k+1,Laux) ;
%p else
%p break ;
%p end if;
%p end do:
%p L1 := DIRICHLET(L1,Laux) ;
%p end if;
%p if p > nops(L1) then
%p break;
%p end if;
%p end do:
%p # this is now Zeta_L(s), seems to be A035187
%p # print(L1) ;
%p # generate Zeta_L(s-1)
%p L1shft := [seq(op(i,L1)*i,i=1..nops(L1))] ;
%p # generate 1/Zeta_L(s)
%p L1x := add(op(i,L1)*x^(i-1),i=1..nops(L1)) :
%p taylor(1/L1x,x=0,nops(L1)) :
%p L1i := gfun[seriestolist](%) ;
%p # generate 1/Zeta_L(2s)
%p L1i2 := [1,seq(0,i=2..nops(L1))] ;
%p for k from 2 to nops(L1i) do
%p if k^2 < nops(L1i2) then
%p L1i2 := subsop(k^2=op(k,L1i),L1i2) ;
%p else
%p break ;
%p end if;
%p end do:
%p # generate Zeta_L(s)*Zeta_L(s-1)
%p DIRICHLET(L1,L1shft) ;
%p # generate Zeta_L(s)*Zeta_L(s-1)/Zeta_L(2s) = Phi(s)
%p Phis := DIRICHLET(%,L1i2) ;
%p # generate Phis(s-1)
%p Phishif := [seq(op(i,Phis)*i,i=1..nops(Phis))] ;
%p DIRICHLET(Phis,Phishif) ;
%Y Cf. A331143.
%K nonn,mult
%O 1,4
%A _N. J. A. Sloane_
%E Terms beyond a(36) from _R. J. Mathar_, Mar 04 2018
%E New name from _Andrey Zabolotskiy_, Feb 16 2021
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