A114438 Number of Barlow packings that repeat after n (or a divisor of n) layers.
0, 1, 1, 2, 1, 4, 3, 8, 8, 18, 21, 48, 63, 133, 205, 412, 685, 1354, 2385, 4644, 8496, 16431, 30735, 59344, 112531, 217246, 415628, 803210, 1545463, 2991192, 5778267, 11201884, 21702708, 42141576, 81830748, 159140896, 309590883, 602938099, 1174779397, 2290920128
Offset: 1
Keywords
Links
- N. J. A. Sloane, Table of n, a(n) for n = 1..500
- Dennis S. Bernstein, Omran Kouba, Counting Colorful Necklaces and Bracelets in Three Colors, arXiv:1901.10703 [math.CO], 2019.
- J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discr. Comp. Geom., 13 (1995), 383-403.
- E. Estevez-Rams, C. Azanza-Ricardo, J. Martínez García and B. Aragón-Fernández, On the algebra of binary codes representing closed-packed staking sequences, Acta Cryst. A61 (2005), 201-208.
- T. J. McLarnan, The numbers of polytypes in close packings and related structures, Zeits. Krist. 155, 269-291 (1981). [See P'(N) on page 272.]
- R. M. Thompson and R. T. Downs, Systematic generation of all nonequivalent closest-packed stacking sequences of length N using group theory, Acta Cryst. B57 (2001), 766-771; B58 (2002), 153.
Programs
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Maple
with(numtheory); read transforms; M:=500; 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; 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; 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
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Mathematica
M = 40; did[m_, n_] := If[Mod[m, n] == 0, 1, 0]; A[n_, d_] := If[Mod[d, 3] == 0, 2^(n/d), (1/3)(2^(n/d) + 2 Cos[Pi n/d])]; 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])]; a[n_] := (1/(4n))(Sum[did[n, d] EulerPhi[d] A[n, d], {d, 1, n}] + EE[n]); Array[a, M] (* Jean-François Alcover, Apr 20 2020, from Maple *)
Comments