A011951 Number of Barlow packings with group P3(bar)m1(S) that repeat after 2n layers.

0, 0, 0, 0, 2, 3, 9, 16, 39, 75, 165, 318, 672, 1323, 2703, 5376, 10880, 21663, 43605, 87040, 174564, 348843, 698709, 1396680, 2795518, 5589675, 11183325, 22364160, 44736512, 89467320, 178951509, 357892096, 715816464, 1431612075, 2863289674, 5726534688, 11453202432

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, eq (24).
• 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:
# eq (25) in Iglesias Z Krist. 221 (2006)
FracR := proc(Phalf)
    if type(Phalf,'even') then
        (bb(Phalf,Phalf)-A045683(Phalf))/2 ;
    else
        0;
    end if;
end proc:
# eq (24) in Iglesias Z Krist. 221 (2006)
A011951 := proc(n)
    local a,p,q,P ;
    P := 2*n ;
    a := FracR(P/2) ;
    for q from 0 to P do
        p := P-q ;
        if modp(p-q,3) = 0 and p < q then
            a := a+bb(p,q) ;
        end if;
    end do:
    a ;
end proc:
seq(A011951(n),n=1..40 ) ; # R. J. Mathar, Apr 15 2024

PARI

apply( {A011951(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\2\d+p\2\d, p\2\d))), bb(p,q)=if(p%2+q%2==0, b(p,q,2)-if((p+q)%4, b(p/2,q/2)))); sum(q=n+1, P, if(q%2==0 && (n-q)*2%3==0, bb(P-q,q)),if(n%2==0,bb(n,n)/2-A045683(n)))/2}, [1..44]) \\ M. F. Hasler, Jun 03 2025

Extensions

More terms from Sean A. Irvine, May 26 2025