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

1, 3, 3, 11, 21, 39, 85, 171, 333, 683, 1364, 2715, 5461, 10923, 21813, 43687, 87381, 174699, 349525, 699051, 1397970, 2796203, 5592402

Offset

3,2

Links

Table of n, a(n) for n=3..25.

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).

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=3..40) ; # R. J. Mathar, Apr 15 2024

Crossrefs

Sequence in context: A278835 A163932 A007022 * A343465 A289883 A280564

Adjacent sequences: A011947 A011948 A011949 * A011951 A011952 A011953

Keyword

nonn,easy

Author

N. J. A. Sloane.

Status

approved