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Number of Barlow packings that repeat after exactly n layers.
3

%I #36 Apr 15 2024 06:45:56

%S 0,1,1,1,1,2,3,6,7,16,21,43,63,129,203,404,685,1343,2385,4625,8492,

%T 16409,30735,59290,112530,217182,415620,803076,1545463,2990968,

%U 5778267,11201472,21702686,42140890,81830744,159139498,309590883,602935713,1174779333,2290915478

%N Number of Barlow packings that repeat after exactly n layers.

%H N. J. A. Sloane, <a href="/A011768/b011768.txt">Table of n, a(n) for n = 1..200</a>

%H Dennis S. Bernstein and 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 E. Estevez-Rams, C. Azanza-Ricardo, J. Martinez-Garcia and B. Argon-Fernandez, <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 E. Esteves-Rams, C. L. Azana Ricardo, B. Aragon Fernandez, <a href="https://doi.org/10.1524/zkri.220.7.592.67101">An alternative expression for counting the number of close-packaged polytypes</a>, Z. Krist. 220 (2005) 592-595, Table 1

%H T. J. McLarnan, <a href="http://dx.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).

%F a(n) = A011946(n/4) + A011947((n-2)/4) + A011948(n/2) + A011949(n/2) + A011950((n+1)/2) + A011951(n/2) + A011952(n/2) + A011953(n) + A011954((n-3)/6) + A011955(n/6-1) + A011955(n/6) + A011956(n/3), where the terms with non-integer indices are set to 0. - _Andrey Zabolotskiy_, Feb 14 2024

%p with(numtheory); read transforms; M:=200;

%p 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 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 PP:=proc(N) (1/(4*N))*(add(did(N,d)*phi(d)*A(N,d), d=1..N)+E(N)); end;

%p for N from 1 to M do t1[N]:=PP(N); od:

%p P:=proc(N) local s,d; s:=0; for d from 1 to N do if N mod d = 0 then s:=s+mobius(N/d)*t1[d]; fi; od: s; end; for N from 1 to M do lprint(N,P(N)); od: # _N. J. A. Sloane_, Aug 10 2006

%t M = 40;

%t did[m_, n_] := If[Mod[m, n] == 0, 1, 0];

%t A[n_, d_] := If[Mod[d, 3] == 0, 2^(n/d), (1/3)(2^(n/d) + 2 Cos[Pi n/d])];

%t EE[n_] := If[Mod[n, 2] == 0, n 2^(n/2) + Sum[did[n/2, d] EulerPhi[2d]* 2^(n/(2 d)), {d, 1, n/2}], (n/3)(2^((n+1)/2) + 2 Cos[Pi(n+1)/2])];

%t PP[n_] := PP[n] = (1/(4n))(Sum[did[n, d] EulerPhi[d] A[n, d], {d, 1, n}] + EE[n]);

%t P[n_] := Module[{s = 0, d}, For[d = 1, d <= n, d++, If[Mod[n, d] == 0, s += MoebiusMu[n/d] PP[d]]]; s];

%t Array[P, M] (* _Jean-François Alcover_, Apr 21 2020, from Maple *)

%Y Cf. A114438.

%Y Cf. A011946, A011947, A011948, A011949, A011950, A011951, A011952, A011953, A011954, A011955, A011956.

%K nonn,easy

%O 1,6

%A _N. J. A. Sloane_ and Michael OKeeffe (MOKeeffe(AT)asu.edu)

%E More terms from _N. J. A. Sloane_, Aug 10 2006