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

1, 2, 4, 9, 19, 40, 80, 165, 330, 672, 1344, 2709, 5418, 10878, 21760, 43605, 87211, 174592, 349180, 698707, 1397418, 2795520, 5591040, 11183436, 22366890, 44736512, 89473020, 178951509, 357903000, 715816960, 1431633920, 2863289683, 5726579370, 11453202383, 22906404864, 45812897109, 91625794218

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• J. E. Iglesias, Enumeration of closest-packings by the space group: a simple approach, Z. Krist. 221 (2006) 237-245.
• T. J. McLarnan, The numbers of polytypes in close packings and related structures, Zeits. Krist. 155, 269-291 (1981).

Programs

Maple

# eq (6) in Iglesias Z Krist. 221 (2006)
b := proc(p,q)
        local d;
        a := 0 ;
        for d from 1 to min(p,q) do
                if modp(p,d)=0 and modp(q,d)=0 then
                        ph := floor(p/2/d) ;
                        qh := floor(q/2/d) ;
                        a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
                end if ;
        end do:
        a ;
end proc:
# eq (17) in Iglesias Z Krist. 221 (2006)
bt := proc(p,q)
        if type(p+q,'odd') then
                b(p,q) ;
        else
                0;
        end if;
end proc:
# corrected eq (15) in Iglesias Z Krist. 221 (2006),  d|(p/2) and d|(q/2)
bbtemp := proc(p,q)
        local d,ph,qh;
        a := 0 ;
        for d from 1 to min(p,q) do
                if modp(p,2*d)=0 and modp(q,2*d)=0 then
                        ph := p/2/d ;
                        qh := q/2/d ;
                        a := a+numtheory[mobius](d)*binomial(ph+qh,ph) ;
                end if ;
        end do:
        a ;
end proc:
# eq (16) in Iglesias Z Krist. 221 (2006)
bb := proc(p,q)
        if type(p,'even') and type(q,'even') then
                ( bbtemp(p,q)-bt(p/2,q/2) )/2 ;
        else
                0 ;
        end if;
end proc:
tt := proc(p,q)
        if type(p+q,'odd') then
                0 ;
        else  # p+q = 2n (below) is always even. - M. F. Hasler, May 27 2025
                b(p,q)-bb(p,q);
        end if;
end proc:
# eq (29) in Iglesias
A011955 := proc(n)
        local a,p,q,P ;
        P := 2*n ;
        a :=0 ;
        for q from 0 to P do
                p := P-q ;
                if modp(p-q,3) <> 0 and p < q then
                        a := a+tt(p,q) ;
                end if;
        end do:
        a ;
end proc:
seq(A011955(n),n=2..40) ;  # R. J. Mathar, Apr 15 2024

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