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

1,2

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

Links

Table of n, a(n) for n=1..34.

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).

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

Example

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

Mathematica

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

Prog

(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])
(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

Crossrefs

Cf. A127804, A000048, A008683.

Cf. A195095 (same sequence).

Sequence in context: A099428 A099920 A128022 * A195095 A331829 A166920

Adjacent sequences: A011943 A011944 A011945 * A011947 A011948 A011949

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Bartosz Naskrecki, Jan 06 2024

Status

approved