cp's OEIS Frontend

A011952 Number of Barlow packings with group P3(bar)m1(O) that repeat after 2n layers.

%I A011952 #13 Jul 08 2025 02:43:30
%S A011952 2,3,8,16,40,73,165,320,666,1323,2709,5364,10880,21675,43577,87040,
%T A011952 174590,348787,698709,1396736,2795400,5589672,11183445,22363920,
%U A011952 44736510,89467563,178951005,357892096,715816960,1431611083,2863289685,5726535661,11453200416
%N A011952 Number of Barlow packings with group P3(bar)m1(O) that repeat after 2n layers.
%H A011952 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 A011952 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 A011952 # eq (6) in Iglesias Z Krist. 221 (2006)
%p A011952 b := proc(p,q)
%p A011952     local d;
%p A011952     a := 0 ;
%p A011952     for d from 1 to min(p,q) do
%p A011952         if modp(p,d)=0 and modp(q,d)=0 then
%p A011952             ph := floor(p/2/d) ;
%p A011952             qh := floor(q/2/d) ;
%p A011952             a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
%p A011952         end if ;
%p A011952     end do:
%p A011952     a ;
%p A011952 end proc:
%p A011952 # eq (17) in Iglesias Z Krist. 221 (2006)
%p A011952 bt := proc(p,q)
%p A011952     if type(p+q,'odd') then
%p A011952         b(p,q) ;
%p A011952     else
%p A011952         0;
%p A011952     end if;
%p A011952 end proc:
%p A011952 # corrected eq (15) in Iglesias Z Krist. 221 (2006),  d|(p/2) and d|(q/2)
%p A011952 bbtemp := proc(p,q)
%p A011952     local d,ph,qh;
%p A011952     a := 0 ;
%p A011952     for d from 1 to min(p,q) do
%p A011952         if modp(p,2*d)=0 and modp(q,2*d)=0 then
%p A011952             ph := p/2/d ;
%p A011952             qh := q/2/d ;
%p A011952             a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
%p A011952         end if ;
%p A011952     end do:
%p A011952     a ;
%p A011952 end proc:
%p A011952 # eq (16) in Iglesias Z Krist. 221 (2006)
%p A011952 bb := proc(p,q)
%p A011952     if type(p,'even') and type(q,'even') then
%p A011952         ( bbtemp(p,q)-bt(p/2,q/2) )/2 ;
%p A011952     else
%p A011952         0 ;
%p A011952     end if;
%p A011952 end proc:
%p A011952 tt := proc(p,q)
%p A011952     if type(p+q,'odd') then
%p A011952         0 ;
%p A011952     else
%p A011952         b(p,q)-bb(p,q);
%p A011952     end if;
%p A011952 end proc:
%p A011952 # eq (28) in Iglesias Z Krist. 221 (2006)
%p A011952 FracS := proc(Phalf)
%p A011952     if type(Phalf,'even') then
%p A011952         tt(Phalf,Phalf)/2 ;
%p A011952     else
%p A011952         (tt(Phalf,Phalf)-A045683(Phalf))/2 ;
%p A011952     end if;
%p A011952 end proc:
%p A011952 # eq (27)
%p A011952 A011952 := proc(n)
%p A011952     local a,p,q,P ;
%p A011952     P := 2*n ;
%p A011952     if type(P,'even') then
%p A011952         a := FracS(P/2) ;
%p A011952         for q from 0 to P do
%p A011952             p := P-q ;
%p A011952             if modp(p-q,3) = 0 and p < q then
%p A011952                 a := a+tt(p,q) ;
%p A011952             end if;
%p A011952         end do:
%p A011952         a ;
%p A011952     else
%p A011952         0;
%p A011952     end if;
%p A011952 end proc:
%p A011952 seq(A011952(n),n=4..40) ;# _R. J. Mathar_, Apr 15 2024
%K A011952 nonn,easy
%O A011952 4,1
%A A011952 _N. J. A. Sloane_
%E A011952 More terms from _Sean A. Irvine_, May 26 2025