

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
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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), 383403.
E. EstevezRams, C. AzanzaRicardo, J. MartinezGarcia and B. ArgonFrenadez, On the algebra of binary codes representing closedpacked staking sequences, Acta Cryst. A61 (2005), 201208.
T. J. McLarnan, The numbers of polytypes in close packings and related structures, Zeits. Krist. 155, 269291 (1981). [See P'(N) on page 272.]
R. M. Thompson and R. T. Downs, Systematic generation of all nonequivalent closestpacked stacking sequences of length N using group theory, Acta Cryst. B57 (2001), 766771; 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


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



