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A114438 Number of Barlow packings that repeat after n (or a divisor of n) layers.

%I A114438 #31 Apr 20 2025 08:05:30
%S A114438 0,1,1,2,1,4,3,8,8,18,21,48,63,133,205,412,685,1354,2385,4644,8496,
%T A114438 16431,30735,59344,112531,217246,415628,803210,1545463,2991192,
%U A114438 5778267,11201884,21702708,42141576,81830748,159140896,309590883,602938099,1174779397,2290920128
%N A114438 Number of Barlow packings that repeat after n (or a divisor of n) layers.
%C A114438 See A011768 for the number of Barlow packings that repeat after exactly n layers.
%C A114438 Like A056353 but with additional restriction that adjacent beads must have different colors.
%H A114438 N. J. A. Sloane, <a href="/A114438/b114438.txt">Table of n, a(n) for n = 1..500</a>
%H A114438 Dennis S. Bernstein, Omran Kouba, <a href="https://arxiv.org/abs/1901.10703">Counting Colorful Necklaces and Bracelets in Three Colors</a>, arXiv:1901.10703 [math.CO], 2019.
%H A114438 J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1007/BF02574051">What are all the best sphere packings in low dimensions?</a>, Discr. Comp. Geom., 13 (1995), 383-403.
%H A114438 E. Estevez-Rams, C. Azanza-Ricardo, J. Martínez García and B. Aragón-Fernández, <a href="https://doi.org/10.1107/S0108767304034294">On the algebra of binary codes representing closed-packed staking sequences</a>, Acta Cryst. A61 (2005), 201-208.
%H A114438 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). [See P'(N) on page 272.]
%H A114438 R. M. Thompson and R. T. Downs, <a href="http://www.geo.arizona.edu/xtal//group/pdf/acB57766.pdf">Systematic generation of all nonequivalent closest-packed stacking sequences of length N using group theory</a>, Acta Cryst. B57 (2001), 766-771; B58 (2002), 153.
%p A114438 with(numtheory); read transforms; M:=500;
%p A114438 A:=proc(N,d) if d mod 3 = 0 then 2^(N/d) else (1/3)*(2^(N/d)+2*cos(Pi*N/d)); fi; end;
%p A114438 E:=proc(N) if N mod 2 = 0 then N*2^(N/2) + add( did(N/2,d)*phi(2*d)*2^(N/(2*d)),d=1..N/2) else (N/3)*(2^((N+1)/2)+2*cos(Pi*(N+1)/2)); fi; end;
%p A114438 PP:=proc(N) (1/(4*N))*(add(did(N,d)*phi(d)*A(N,d), d=1..N)+E(N)); end; for N from 1 to M do lprint(N,PP(N)); od: # _N. J. A. Sloane_, Aug 10 2006
%t A114438 M = 40;
%t A114438 did[m_, n_] := If[Mod[m, n] == 0, 1, 0];
%t A114438 A[n_, d_] := If[Mod[d, 3] == 0, 2^(n/d), (1/3)(2^(n/d) + 2 Cos[Pi n/d])];
%t A114438 EE[n_] := If[Mod[n, 2] == 0, n 2^(n/2) + Sum[did[n/2, d] EulerPhi[2d] 2^(n/(2d)), {d, 1, n/2}], (n/3)(2^((n+1)/2) + 2 Cos[Pi(n+1)/2])];
%t A114438 a[n_] := (1/(4n))(Sum[did[n, d] EulerPhi[d] A[n, d], {d, 1, n}] + EE[n]);
%t A114438 Array[a, M] (* _Jean-François Alcover_, Apr 20 2020, from Maple *)
%Y A114438 Cf. A027671, A056353.
%K A114438 nonn
%O A114438 1,4
%A A114438 _N. J. A. Sloane_, Feb 28 2006; more terms, Aug 10 2006