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A114438 Number of Barlow packings that repeat after n (or a divisor of n) layers. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

See A011768 for the number of Barlow packings that repeat after exactly n layers.

Like A056353 but with additional restriction that adjacent beads must have different colors.

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. Martinez-Garcia and B. Argon-Frenadez, 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.

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

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-Fran├žois Alcover, Apr 20 2020, from Maple *)

CROSSREFS

Cf. A027671, A056353.

Sequence in context: A106624 A028297 A207537 * A238757 A306888 A262230

Adjacent sequences:  A114435 A114436 A114437 * A114439 A114440 A114441

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 28 2006; more terms, Aug 10 2006

STATUS

approved

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Last modified July 7 14:34 EDT 2020. Contains 335495 sequences. (Running on oeis4.)