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%I #12 Apr 15 2024 11:55:15
%S 1,2,4,9,19,40,80,165,330,672,1344,2709,5418,10878,21760,43605,87211,
%T 174592,349180,698707,1397418,2795520,5591040,11183436,22366890,
%U 44736512,89473020,178951509,357903000,715816960,1431633920,2863289683,5726579370,11453202383,22906404864,45812897109,91625794218
%N Number of Barlow packings with group R3(bar)m(O) that repeat after 6n layers.
%H J. E. Iglesias, <a href="https://doi.org/10.1524/zkri.2006.221.4.237">Enumeration of closest-packings by the space group: a simple approach</a>, Z. Krist. 221 (2006) 237-245.
%H T. J. McLarnan, <a href="http://dx.doi.org/10.1524/zkri.1981.155.3-4.269">The numbers of polytypes in close packings and related structures</a>, Zeits. Krist. 155, 269-291 (1981).
%p # eq (6) in Iglesias Z Krist. 221 (2006)
%p b := proc(p,q)
%p local d;
%p a := 0 ;
%p for d from 1 to min(p,q) do
%p if modp(p,d)=0 and modp(q,d)=0 then
%p ph := floor(p/2/d) ;
%p qh := floor(q/2/d) ;
%p a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
%p end if ;
%p end do:
%p a ;
%p end proc:
%p # eq (17) in Iglesias Z Krist. 221 (2006)
%p bt := proc(p,q)
%p if type(p+q,'odd') then
%p b(p,q) ;
%p else
%p 0;
%p end if;
%p end proc:
%p # corrected eq (15) in Iglesias Z Krist. 221 (2006), d|(p/2) and d|(q/2)
%p bbtemp := proc(p,q)
%p local d,ph,qh;
%p a := 0 ;
%p for d from 1 to min(p,q) do
%p if modp(p,2*d)=0 and modp(q,2*d)=0 then
%p ph := p/2/d ;
%p qh := q/2/d ;
%p a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
%p end if ;
%p end do:
%p a ;
%p end proc:
%p # eq (16) in Iglesias Z Krist. 221 (2006)
%p bb := proc(p,q)
%p if type(p,'even') and type(q,'even') then
%p ( bbtemp(p,q)-bt(p/2,q/2) )/2 ;
%p else
%p 0 ;
%p end if;
%p end proc:
%p tt := proc(p,q)
%p if type(p+q,'odd') then
%p 0 ;
%p else
%p b(p,q)-bb(p,q);
%p end if;
%p end proc:
%p # eq (29) in Iglesias
%p A011955 := proc(n)
%p local a,p,q,P ;
%p P := 2*n ;
%p a :=0 ;
%p for q from 0 to P do
%p p := P-q ;
%p if modp(p-q,3) <> 0 and p < q then
%p a := a+tt(p,q) ;
%p end if;
%p end do:
%p a ;
%p end proc:
%p seq(A011955(n),n=2..40) ; # _R. J. Mathar_, Apr 15 2024
%K nonn,easy
%O 2,2
%A _N. J. A. Sloane_.
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