A011956 - OEIS

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

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OFFSET

7,2

COMMENTS

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

LINKS

Table of n, a(n) for n=7..40.

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.

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

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 *)

PROG

(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

CROSSREFS

Cf. A011768, A011954, A011955, A371992.

Sequence in context: A175204 A215533 A128461 * A371358 A018003 A328692

Adjacent sequences: A011953 A011954 A011955 * A011957 A011958 A011959