A011951 - OEIS

%I #9 Apr 15 2024 12:17:01

%S 2,3,9,16,39,75,165,318,672,1323,2703,5376,10880,21663,43605,87040,

%T 174564,348843,698709,1396680,2795518

%N Number of Barlow packings with group P3(bar)m1(S) that repeat after 2n 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, eq (24).

%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 # eq (25) in Iglesias Z Krist. 221 (2006)

%p FracR := proc(Phalf)

%p     if type(Phalf,'even') then

%p         (bb(Phalf,Phalf)-A045683(Phalf))/2 ;

%p     else

%p         0;

%p     end if;

%p end proc:

%p # eq (24) in Iglesias Z Krist. 221 (2006)

%p A011951 := proc(n)

%p     local a,p,q,P ;

%p     P := n ;

%p     a := FracR(P/2) ;

%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+bb(p,q) ;

%p         end if;

%p     end do:

%p     a ;

%p end proc:

%p seq(A011951(2*n),n=5..40 ) ; # _R. J. Mathar_, Apr 15 2024

%K nonn,easy

%O 5,1

%A _N. J. A. Sloane_.