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
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Bartosz Naskrecki, Jan 06 2024
STATUS
approved