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%I #43 Apr 15 2024 07:30:33
%S 1,2,3,8,15,30,63,128,252,510,1023,2040,4095,8190,16365,32768,65535,
%T 131040,262143,524280,1048509,2097150,4194303,8388480,16777200,
%U 33554430,67108608,134217720,268435455,536870370,1073741823,2147483648,4294966269,8589934590
%N Number of Barlow packings with group P63/mmc(S) that repeat after 4n layers.
%C Conjecture: Mobius transform of A127804. - _R. J. Mathar_, Sep 14 2011
%C 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
%H T. J. McLarnan, <a href="https://doi.org/10.1524/zkri.1981.155.3-4.269">The numbers of polytypes in close packings and related structures</a>, Zeits. Krist. 155, 269-291 (1981).
%H Juan E. Iglesias, <a href="https://doi.org/10.1524/zkri.2006.221.4.237">Enumeration of closest-packings by the space group: a simple approach</a>, Zeits. Krist., 221(4), 237-245 (2006).
%F From _Paul D. Hanna_, Sep 08 2011 (merged from duplicate A195095 Feb 06 2024): (Start)
%F G.f.: Sum_{n>=1} -moebius(2*n)*x^n/(1 - 2*x^n).
%F a(2^n) = 2^(2^n - 1).
%F a(p) = 2^(p-1) for odd prime p.
%F (End)
%F a(n) = floor(2^(n-1)/n)*n unless n=3k, k>4. - _M. F. Hasler_, Sep 08 2011
%F G.f.: Sum_{n>=1} moebius(2*n-1)*x^(2*n-1)/(1 - 2*x^(2*n-1)). - _Mamuka Jibladze_, Dec 04 2016
%F 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
%F Conjecture: a(n) = A045683(2n). - _R. J. Mathar_, Apr 15 2024
%e G.f. = x + 2*x^2 + 3*x^3 + 8*x^4 + 15*x^5 + 30*x^6 + 63*x^7 + 128*x^8 + ...
%t a[n_] := DivisorSum[n, MoebiusMu[#] * 2^(n/#-1) &, OddQ[#] &]; Array[a, 35] (* _Amiram Eldar_, Aug 29 2023 *)
%o (Python)
%o 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])
%o (PARI) {a(n)=polcoeff(sum(m=1,n,-moebius(2*m)*x^m/(1-2*x^m+x*O(x^n))^1),n)}
%o for(n=1,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 08 2011
%o (PARI) {a(n) = sumdiv(n,d, if(d%2==1, moebius(d) * 2^(n/d-1) ) )}
%o for(n=1,30,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 08 2011
%Y Cf. A127804, A000048, A008683.
%Y Cf. A195095 (same sequence).
%K nonn,easy,changed
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Bartosz Naskrecki_, Jan 06 2024
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