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A011950 Number of Barlow packings with group P3(bar)m1(SO) that repeat after 2n-1 layers.

%I A011950 #20 Jul 08 2025 02:43:17
%S A011950 0,0,1,3,3,11,21,39,85,171,333,683,1364,2715,5461,10923,21813,43687,
%T A011950 87381,174699,349525,699051,1397970,2796203,5592402,11184555,22369621,
%U A011950 44739231,89477973,178956971,357913941,715826856,1431655743,2863311531,5726621013,11453246123
%N A011950 Number of Barlow packings with group P3(bar)m1(SO) that repeat after 2n-1 layers.
%H A011950 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 A011950 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 A011950 # eq (6) in Iglesias Z Krist. 221 (2006)
%p A011950 b := proc(p,q)
%p A011950     local d;
%p A011950     a := 0 ;
%p A011950     for d from 1 to min(p,q) do
%p A011950         if modp(p,d)=0 and modp(q,d)=0 then
%p A011950             ph := floor(p/2/d) ;
%p A011950             qh := floor(q/2/d) ;
%p A011950             a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
%p A011950         end if ;
%p A011950     end do:
%p A011950     a ;
%p A011950 end proc:
%p A011950 # eq (17) in Iglesias Z Krist. 221 (2006)
%p A011950 bt := proc(p,q)
%p A011950     if type(p+q,'odd') then
%p A011950         b(p,q) ;
%p A011950     else
%p A011950         0;
%p A011950     end if;
%p A011950 end proc:
%p A011950 # eq (30) in Iglesias Z Krist. 221 (2006)
%p A011950 A011950 := proc(n)
%p A011950     local a, P, p, q ;
%p A011950     if n = 0 then
%p A011950         1;
%p A011950     else
%p A011950         P := 2*n-1 ;
%p A011950         a :=0 ;
%p A011950         for q from 0 to P do
%p A011950             p := P-q ;
%p A011950             if modp(p-q, 3) = 0 and p <= q then
%p A011950                 a := a+bt(p, q) ;
%p A011950             end if;
%p A011950         end do:
%p A011950         a ;
%p A011950     end if;
%p A011950 end proc:
%p A011950 seq(A011950(n), n=1..40) ; # _R. J. Mathar_, Apr 15 2024
%o A011950 (PARI) apply( {A011950(n)=my(P=2*n-1, b(p, q)=sum(d=1, min(p, q), if(p%d+q%d==0, moebius(d)*binomial(q\2\d+p\2\d, p\2\d)))); sum(q=n+1,P, if(2*(n-q)%3==1, b(P-q,q)))}, [1..44])
%K A011950 nonn,easy
%O A011950 1,4
%A A011950 _N. J. A. Sloane_
%E A011950 More terms from _Sean A. Irvine_, May 26 2025
%E A011950 Offset changed to 1 and a(1) = a(2) = 0 included by _M. F. Hasler_, May 28 2025