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

0, 1, 1, 1, 1, 2, 3, 6, 7, 16, 21, 43, 63, 129, 203, 404, 685, 1343, 2385, 4625, 8492, 16409, 30735, 59290, 112530, 217182, 415620, 803076, 1545463, 2990968, 5778267, 11201472, 21702686, 42140890, 81830744, 159139498, 309590883, 602935713, 1174779333, 2290915478, 4469734225, 8726815264

Offset: 1

Author

N. J. A. Sloane and Michael OKeeffe (MOKeeffe(AT)asu.edu)

Keywords

• nonn
• easy

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..200
• Dennis S. Bernstein and Omran Kouba, Counting Colorful Necklaces and Bracelets in Three Colors, arXiv:1901.10703 [math.CO], 2019.
• 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.
• Ernesto Estevez-Rams, Cristy Azanza Ricardo and Beatriz Aragón Fernández, An alternative expression for counting the number of close-packaged polytypes, Z. Krist. 220 (2005) 592-595, Table 1
• T. J. McLarnan, The numbers of polytypes in close-packings and related structures, Zeits. Krist. 155, 269-291 (1981).

Crossrefs

Cf. A114438, A371991, A371992.

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

Programs

Maple

with(numtheory); read transforms; M:=200;
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 t1[N]:=PP(N); od:
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

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/(2 d)), {d, 1, n/2}], (n/3)(2^((n+1)/2) + 2 Cos[Pi(n+1)/2])];
PP[n_] := PP[n] = (1/(4n))(Sum[did[n, d] EulerPhi[d] A[n, d], {d, 1, n}] + EE[n]);
P[n_] := Module[{s = 0, d}, For[d = 1, d <= n, d++, If[Mod[n, d] == 0, s += MoebiusMu[n/d] PP[d]]]; s];
Array[P, M] (* Jean-François Alcover, Apr 21 2020, from Maple *)

PARI

apply( {A011768(n)=A371991(n)+if(n%3, 0, n>3, A371992(n/3), 1)}, [1..42]) \\ M. F. Hasler, May 27 2025

Formula

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

Extensions

More terms from N. J. A. Sloane, Aug 10 2006