A011954 Barlow packings with group R3(bar)m(SO) that repeat after 6n+3 layers.

1, 1, 2, 4, 11, 20, 42, 84, 170, 340, 682, 1364, 2728, 5461, 10922, 21844, 43690, 87374, 174762, 349524, 699050, 1398100, 2796192, 5592404, 11184806, 22369620, 44739242, 89478462, 178956970, 357913940, 715827882, 1431655754, 2863311486, 5726623060, 11453246122, 22906492244, 45812984490

Offset: 0

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;  # never happens since p+q = P = 2n+1. - M. F. Hasler, May 27 2025
    end if;
end proc:
# eq (31) in Iglesias Z Krist. 221 (2006)
A011954 := 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(A011954(n),n=0..40) ; # R. J. Mathar, Apr 15 2024

PARI

apply( {A011954(n)=if(n, my(P=2*n+1, b(p,q)=sum(d=1, min(p,q), if(!(p%d || q%d), moebius(d)*binomial(q\d\2+p\d\2, p\d\2)))); sum(q=P\/2,P, if((P-q*2)%3, b(P-q, q))),1)}, [0..33]) \\ M. F. Hasler, May 27 2025