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

0, 0, 1, 3, 3, 11, 21, 39, 85, 171, 333, 683, 1364, 2715, 5461, 10923, 21813, 43687, 87381, 174699, 349525, 699051, 1397970, 2796203, 5592402, 11184555, 22369621, 44739231, 89477973, 178956971, 357913941, 715826856, 1431655743, 2863311531, 5726621013, 11453246123

Offset: 1

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:
# eq (30) in Iglesias Z Krist. 221 (2006)
A011950 := proc(n)
    local a, P, p, q ;
    if n = 0 then
        1;
    else
        P := 2*n-1 ;
        a :=0 ;
        for q from 0 to P do
            p := P-q ;
            if modp(p-q, 3) = 0 and p <= q then
                a := a+bt(p, q) ;
            end if;
        end do:
        a ;
    end if;
end proc:
seq(A011950(n), n=1..40) ; # R. J. Mathar, Apr 15 2024

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])

Extensions

More terms from Sean A. Irvine, May 26 2025

Offset changed to 1 and a(1) = a(2) = 0 included by M. F. Hasler, May 28 2025