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

2,2

Links

Table of n, a(n) for n=2..38.

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:
# 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
                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

Crossrefs

Sequence in context: A099568 A018001 A018099 * A084172 A018100 A052908

Adjacent sequences: A011952 A011953 A011954 * A011956 A011957 A011958

Keyword

nonn,easy

Author

N. J. A. Sloane.

Status

approved