A011946 Number of Barlow packings with group P63/mmc(S) that repeat after 4n layers.

1, 2, 3, 8, 15, 30, 63, 128, 252, 510, 1023, 2040, 4095, 8190, 16365, 32768, 65535, 131040, 262143, 524280, 1048509, 2097150, 4194303, 8388480, 16777200, 33554430, 67108608, 134217720, 268435455, 536870370, 1073741823, 2147483648, 4294966269, 8589934590

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Conjecture: Mobius transform of A127804. - R. J. Mathar, Sep 14 2011

a(n) = n*A000048(n), where A000048(n) = number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged. - Paul D. Hanna, Dec 21 2016

Examples

			G.f. = x + 2*x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 30*x^6 + 63*x^7 + 128*x^8 + ...
		

Links

• T. J. McLarnan, The numbers of polytypes in close packings and related structures, Zeits. Krist. 155, 269-291 (1981).
• Juan E. Iglesias, Enumeration of closest-packings by the space group: a simple approach, Zeits. Krist., 221(4), 237-245 (2006).

Crossrefs

Cf. A127804, A000048, A008683.

Cf. A195095 (same sequence).

Programs

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[#] * 2^(n/#-1) &, OddQ[#] &]; Array[a, 35] (* Amiram Eldar, Aug 29 2023 *)

PARI

{a(n)=polcoeff(sum(m=1,n,-moebius(2*m)*x^m/(1-2*x^m+x*O(x^n))^1),n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Sep 08 2011

PARI

{a(n) =  sumdiv(n,d, if(d%2==1, moebius(d) * 2^(n/d-1) ) )}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Sep 08 2011

Python

a = lambda n: sum([sympy.mobius(d)*(2**(sympy.floor((2*n//d+1)/2)-1)) for d in sympy.divisors(n) if d%2==1])

Formula

From Paul D. Hanna, Sep 08 2011 (merged from duplicate A195095 Feb 06 2024): (Start)

G.f.: Sum_{n>=1} -moebius(2*n)*x^n/(1 - 2*x^n).

a(2^n) = 2^(2^n - 1).

a(p) = 2^(p-1) for odd prime p.

(End)

a(n) = floor(2^(n-1)/n)*n unless n=3k, k>4. - M. F. Hasler, Sep 08 2011

G.f.: Sum_{n>=1} moebius(2*n-1)*x^(2*n-1)/(1 - 2*x^(2*n-1)). - Mamuka Jibladze, Dec 04 2016

a(n) = Sum_{d|n, d odd} moebius(d) * 2^(n/d - 1), where moebius(n) = A008683(n). - Paul D. Hanna, Dec 21 2016, Bartosz Naskrecki, Jan 06 2024

Conjecture: a(n) = A045683(2n). - R. J. Mathar, Apr 15 2024

Extensions

More terms from Bartosz Naskrecki, Jan 06 2024