A011956 Number of close-packings with layer-number 3n and space group R3m.

1, 2, 4, 10, 21, 42, 84, 164, 322, 620, 1200, 2300, 4429, 8482, 16303, 31259, 60105, 115472, 222332, 428106, 825774, 1593669, 3080004, 5956902, 11534689, 22352962, 43361663, 84181720, 163574114, 318079104, 619007004, 1205471654, 2349209058, 4581032192

Offset: 7

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Last column of Table 4 in McLarnan (1981), p. 277. See there for more information. - M. F. Hasler, May 26 2025

Links

• Juan E. Iglesias, Enumeration of closest-packings by the space group: a simple approach, Z. Krist. 221 (2006) 237-245. See Table 4.
• T. J. McLarnan, The numbers of polytypes in close-packings and related structures, Zeits. Krist. 155, 269-291 (1981).
• Eric W. Weisstein, Barlow Packing, on MathWorld-A Wolfram Web Resource.
• Wikipedia, Close-packing of equal spheres.

Crossrefs

Cf. A011768, A011954, A011955, A371992.

Programs

Mathematica

fa[p_, q_] := fa[p, q] = (p+q-1)!/(p!q!) - Sum[fa[p/d, q/d]/d, {d, Rest[Intersection@@(Divisors/@{p, q})]}];
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})]}];
rh[n_] := Sum[fa[n-q, q]+fb[n-q, q], {q, Select[Range[n/2], !Divisible[n-2#, 3]&]}] / 2; (* A371992 *)
fSO[n_] := Sum[fb[2n+1-q,q], {q, Select[Range[n+1,2n], !Divisible[2n+1-2#,3]&]}];(*A011954*)
fb2[p_, q_] := fb2[p, q] = (p+q)!/(p!q!) - Sum[fb2[p/d, q/d], {d, Rest[Intersection@@(Divisors/@{p, q})]}]; (*A050186(p+q, p)*)
fO[n_] := Sum[fb[2n-q, q] - If[EvenQ@q, fb2[n-q/2, q/2] - If[OddQ@n, fb[n-q/2, q/2], 0], 0] / 2, {q, Select[Range[n+1, 2n-1], !Divisible[n-#, 3]&]}]; (*A011955*)
a[n_] := rh[n] - If[OddQ@n, fSO[(n-1)/2], fO[n/2]+fO[n/2-1]];
Table[a[n],{n,7,50}] (* Andrei Zabolotskii, May 30 2025 *)

PARI

apply( {A011956(n) = A371992(n) - if(n%2,A011954(n\2), A011955(n/2)+A011955(n/2-1))}, [7..20]) \\ M. F. Hasler, May 27 2025

Python

def A011956(n): return A371992(n) - (A011954(n//2) if n&1 else A011955(n//2)+A011955(n//2-1))
# M. F. Hasler, May 27 2025

Formula

a(n) = A371992(n) - A011954((n-1)/2) - A011955(n/2) - A011955(n/2-1), where the terms with non-integer indices are set to 0. - Andrei Zabolotskii and M. F. Hasler, May 27 2025

Extensions

Name and offset corrected by Andrei Zabolotskii, Feb 14 2024

Name changed by M. F. Hasler, May 26 2025

Terms a(17) onwards from Andrei Zabolotskii, May 30 2025