A371991 -id:A371991 - cp's OEIS frontend

A371992 Number of different closest packings of equal spheres for rhombohedral crystals having repeat period n.

0, 0, 1, 1, 2, 3, 5, 8, 15, 23, 41, 70, 126, 223, 406, 740, 1370, 2545, 4769, 8977, 16985, 32261, 61469, 117488, 225060, 432159, 831235, 1601796, 3090926, 5973198, 11556533, 22385600, 43405353, 84247085, 163661488, 318209920, 619181766, 1205733457, 2349558582, 4581555964, 8939468450, 17453081143, 34094082857

Offset: 1

R. J. Mathar, Apr 15 2024

nonn

J. E. Iglesias, A formula for the number of closest packings of equal spheres having a given repeat period, Z. Krist. 155 (1981) 121-127, Table 1.

Cf. A371991, A000046, A051168, A180424, A000740, A000048.

fa[p_,q_] := fa[p,q] = (p+q-1)!/(p!q!) - Sum[fa[p/d,q/d]/d, {d, Rest[Intersection@@(Divisors/@{p,q})]}]; (*A051168(p+q,p); Iglesias Eq. (1)*)
fb[p_,q_] := fb[p,q] = (Quotient[p,2]+Quotient[q,2])!/(Quotient[p,2]!Quotient[q,2]!) - Sum[fb[p/d,q/d], {d, Rest[Intersection@@(Divisors/@{p,q})]}]; (*A180424(p+q,p); Eq. (2)*)
am[p_] := am[p] = 2^(p-1) - Sum[am[p/d], {d, Rest@Divisors@p}]; (*A000740; Eq. (6)*)
atf[p_] := atf[p] = 2^(p-1)/p - Sum[atf[p/d]/d, {d, Select[Rest@Divisors@p, OddQ]}];(*A000048; Eq. (9)*)
a[n_] := Sum[With[{p=n-q}, fa[p,q]+fb[p,q] + If[p==q, am[p]+atf[p]-fa[p,q]-fb[p,q], 0] / 2], {q, Select[Range[n/2], !Divisible[n-2#,3]& (*the opposite condition would give A371991*)]}] / 2; (* Eq. (5) *)
Table[a[n], {n, 2, 40}] (* Andrei Zabolotskii, May 30 2025 *)

apply( {A371992(n)=sum(q=1, n\2, if((n-2*q)%3, A051168(n,q)+A180424(n,q)))/2}, [1..40]) \\ M. F. Hasler, Jun 05 2025

a(n) + A371991(n) = A000046(n).

a(n+1)/a(n) = 2 - 2/n + o(1/n). - M. F. Hasler, Jun 09 2025

Offset changed to 1 and a(1) = 0 prefixed by M. F. Hasler, Jun 05 2025

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

Original entry on oeis.org

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

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

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).

Cf. A114438, A371991, A371992.

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

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

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 *)

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

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

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

cp's OEIS Frontend

A371992 Number of different closest packings of equal spheres for rhombohedral crystals having repeat period n.

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