cp's OEIS Frontend

A011955 Number of Barlow packings with group R3(bar)m(O) that repeat after 6n layers.

%I A011955 #19 Jul 08 2025 02:43:49
%S A011955 1,2,4,9,19,40,80,165,330,672,1344,2709,5418,10878,21760,43605,87211,
%T A011955 174592,349180,698707,1397418,2795520,5591040,11183436,22366890,
%U A011955 44736512,89473020,178951509,357903000,715816960,1431633920,2863289683,5726579370,11453202383,22906404864,45812897109,91625794218
%N A011955 Number of Barlow packings with group R3(bar)m(O) that repeat after 6n layers.
%H A011955 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 A011955 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 A011955 # eq (6) in Iglesias Z Krist. 221 (2006)
%p A011955 b := proc(p,q)
%p A011955         local d;
%p A011955         a := 0 ;
%p A011955         for d from 1 to min(p,q) do
%p A011955                 if modp(p,d)=0 and modp(q,d)=0 then
%p A011955                         ph := floor(p/2/d) ;
%p A011955                         qh := floor(q/2/d) ;
%p A011955                         a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
%p A011955                 end if ;
%p A011955         end do:
%p A011955         a ;
%p A011955 end proc:
%p A011955 # eq (17) in Iglesias Z Krist. 221 (2006)
%p A011955 bt := proc(p,q)
%p A011955         if type(p+q,'odd') then
%p A011955                 b(p,q) ;
%p A011955         else
%p A011955                 0;
%p A011955         end if;
%p A011955 end proc:
%p A011955 # corrected eq (15) in Iglesias Z Krist. 221 (2006),  d|(p/2) and d|(q/2)
%p A011955 bbtemp := proc(p,q)
%p A011955         local d,ph,qh;
%p A011955         a := 0 ;
%p A011955         for d from 1 to min(p,q) do
%p A011955                 if modp(p,2*d)=0 and modp(q,2*d)=0 then
%p A011955                         ph := p/2/d ;
%p A011955                         qh := q/2/d ;
%p A011955                         a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
%p A011955                 end if ;
%p A011955         end do:
%p A011955         a ;
%p A011955 end proc:
%p A011955 # eq (16) in Iglesias Z Krist. 221 (2006)
%p A011955 bb := proc(p,q)
%p A011955         if type(p,'even') and type(q,'even') then
%p A011955                 ( bbtemp(p,q)-bt(p/2,q/2) )/2 ;
%p A011955         else
%p A011955                 0 ;
%p A011955         end if;
%p A011955 end proc:
%p A011955 tt := proc(p,q)
%p A011955         if type(p+q,'odd') then
%p A011955                 0 ;
%p A011955         else  # p+q = 2n (below) is always even. - _M. F. Hasler_, May 27 2025
%p A011955                 b(p,q)-bb(p,q);
%p A011955         end if;
%p A011955 end proc:
%p A011955 # eq (29) in Iglesias
%p A011955 A011955 := proc(n)
%p A011955         local a,p,q,P ;
%p A011955         P := 2*n ;
%p A011955         a :=0 ;
%p A011955         for q from 0 to P do
%p A011955                 p := P-q ;
%p A011955                 if modp(p-q,3) <> 0 and p < q then
%p A011955                         a := a+tt(p,q) ;
%p A011955                 end if;
%p A011955         end do:
%p A011955         a ;
%p A011955 end proc:
%p A011955 seq(A011955(n),n=2..40) ;  # _R. J. Mathar_, Apr 15 2024
%o A011955 (PARI) apply( {A011955(n)=my(P=2*n, b(p, q, f=1)=sum(d=1, min(p, q), if(p%(d*f)+q%(d*f)==0, moebius(d)*binomial(q\d\2+p\d\2, p\d\2)))); sum(q=n+1, 2*n, if(2*(n-q)%3, b(2*n-q, q)-if(q%2==0, b(2*n-q, q, 2)-if(n%2,b(n-q/2,q/2)))/2))}, [2..35]) \\ _M. F. Hasler_, May 27 2025
%K A011955 nonn,easy
%O A011955 2,2
%A A011955 _N. J. A. Sloane_