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A011951 Number of Barlow packings with group P3(bar)m1(S) that repeat after 2n layers.

%I A011951 #21 Jul 08 2025 02:43:24
%S A011951 0,0,0,0,2,3,9,16,39,75,165,318,672,1323,2703,5376,10880,21663,43605,
%T A011951 87040,174564,348843,698709,1396680,2795518,5589675,11183325,22364160,
%U A011951 44736512,89467320,178951509,357892096,715816464,1431612075,2863289674,5726534688,11453202432
%N A011951 Number of Barlow packings with group P3(bar)m1(S) that repeat after 2n layers.
%H A011951 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 A011951 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 A011951 # eq (6) in Iglesias Z Krist. 221 (2006)
%p A011951 b := proc(p,q)
%p A011951     local d;
%p A011951     a := 0 ;
%p A011951     for d from 1 to min(p,q) do
%p A011951         if modp(p,d)=0 and modp(q,d)=0 then
%p A011951             ph := floor(p/2/d) ;
%p A011951             qh := floor(q/2/d) ;
%p A011951             a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
%p A011951         end if ;
%p A011951     end do:
%p A011951     a ;
%p A011951 end proc:
%p A011951 # eq (17) in Iglesias Z Krist. 221 (2006)
%p A011951 bt := proc(p,q)
%p A011951     if type(p+q,'odd') then
%p A011951         b(p,q) ;
%p A011951     else
%p A011951         0;
%p A011951     end if;
%p A011951 end proc:
%p A011951 # corrected eq (15) in Iglesias Z Krist. 221 (2006),  d|(p/2) and d|(q/2)
%p A011951 bbtemp := proc(p,q)
%p A011951     local d,ph,qh;
%p A011951     a := 0 ;
%p A011951     for d from 1 to min(p,q) do
%p A011951         if modp(p,2*d)=0 and modp(q,2*d)=0 then
%p A011951             ph := p/2/d ;
%p A011951             qh := q/2/d ;
%p A011951             a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
%p A011951         end if ;
%p A011951     end do:
%p A011951     a ;
%p A011951 end proc:
%p A011951 # eq (16) in Iglesias Z Krist. 221 (2006)
%p A011951 bb := proc(p,q)
%p A011951     if type(p,'even') and type(q,'even') then
%p A011951         ( bbtemp(p,q)-bt(p/2,q/2) )/2 ;
%p A011951     else
%p A011951         0 ;
%p A011951     end if;
%p A011951 end proc:
%p A011951 # eq (25) in Iglesias Z Krist. 221 (2006)
%p A011951 FracR := proc(Phalf)
%p A011951     if type(Phalf,'even') then
%p A011951         (bb(Phalf,Phalf)-A045683(Phalf))/2 ;
%p A011951     else
%p A011951         0;
%p A011951     end if;
%p A011951 end proc:
%p A011951 # eq (24) in Iglesias Z Krist. 221 (2006)
%p A011951 A011951 := proc(n)
%p A011951     local a,p,q,P ;
%p A011951     P := 2*n ;
%p A011951     a := FracR(P/2) ;
%p A011951     for q from 0 to P do
%p A011951         p := P-q ;
%p A011951         if modp(p-q,3) = 0 and p < q then
%p A011951             a := a+bb(p,q) ;
%p A011951         end if;
%p A011951     end do:
%p A011951     a ;
%p A011951 end proc:
%p A011951 seq(A011951(n),n=1..40 ) ; # _R. J. Mathar_, Apr 15 2024
%o A011951 (PARI) apply( {A011951(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\2\d+p\2\d, p\2\d))), bb(p,q)=if(p%2+q%2==0, b(p,q,2)-if((p+q)%4, b(p/2,q/2)))); sum(q=n+1, P, if(q%2==0 && (n-q)*2%3==0, bb(P-q,q)),if(n%2==0,bb(n,n)/2-A045683(n)))/2}, [1..44]) \\ _M. F. Hasler_, Jun 03 2025
%K A011951 nonn,easy
%O A011951 1,5
%A A011951 _N. J. A. Sloane_
%E A011951 More terms from _Sean A. Irvine_, May 26 2025