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A011768 Number of Barlow packings that repeat after exactly n layers.

%I A011768 #44 May 28 2025 01:10:42
%S A011768 0,1,1,1,1,2,3,6,7,16,21,43,63,129,203,404,685,1343,2385,4625,8492,
%T A011768 16409,30735,59290,112530,217182,415620,803076,1545463,2990968,
%U A011768 5778267,11201472,21702686,42140890,81830744,159139498,309590883,602935713,1174779333,2290915478,4469734225,8726815264
%N A011768 Number of Barlow packings that repeat after exactly n layers.
%H A011768 N. J. A. Sloane, <a href="/A011768/b011768.txt">Table of n, a(n) for n = 1..200</a>
%H A011768 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 A011768 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 A011768 Ernesto Estevez-Rams, Cristy Azanza Ricardo and Beatriz Aragón Fernández, <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 A011768 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).
%F A011768 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. For n > 3, the two parenthesized terms are resp. A371991(n) and A371992(n/3). - _Andrey Zabolotskiy_, Feb 14 2024 and May 27 2025
%p A011768 with(numtheory); read transforms; M:=200;
%p A011768 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 A011768 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 A011768 PP:=proc(N) (1/(4*N))*(add(did(N,d)*phi(d)*A(N,d), d=1..N)+E(N)); end;
%p A011768 for N from 1 to M do t1[N]:=PP(N); od:
%p A011768 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 A011768 M = 40;
%t A011768 did[m_, n_] := If[Mod[m, n] == 0, 1, 0];
%t A011768 A[n_, d_] := If[Mod[d, 3] == 0, 2^(n/d), (1/3)(2^(n/d) + 2 Cos[Pi n/d])];
%t A011768 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 A011768 PP[n_] := PP[n] = (1/(4n))(Sum[did[n, d] EulerPhi[d] A[n, d], {d, 1, n}] + EE[n]);
%t A011768 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 A011768 Array[P, M] (* _Jean-François Alcover_, Apr 21 2020, from Maple *)
%o A011768 (PARI) apply( {A011768(n)=A371991(n)+if(n%3, 0, n>3, A371992(n/3), 1)}, [1..42]) \\ _M. F. Hasler_, May 27 2025
%Y A011768 Cf. A114438, A371991, A371992.
%Y A011768 Cf. A011946, A011947, A011948, A011949, A011950, A011951, A011952, A011953, A011954, A011955, A011956.
%K A011768 nonn,easy
%O A011768 1,6
%A A011768 _N. J. A. Sloane_ and Michael OKeeffe (MOKeeffe(AT)asu.edu)
%E A011768 More terms from _N. J. A. Sloane_, Aug 10 2006