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

%I A011956 #30 Jun 18 2025 00:56:13
%S A011956 1,2,4,10,21,42,84,164,322,620,1200,2300,4429,8482,16303,31259,60105,
%T A011956 115472,222332,428106,825774,1593669,3080004,5956902,11534689,
%U A011956 22352962,43361663,84181720,163574114,318079104,619007004,1205471654,2349209058,4581032192
%N A011956 Number of close-packings with layer-number 3n and space group R3m.
%C A011956 Last column of Table 4 in McLarnan (1981), p. 277. See there for more information. - _M. F. Hasler_, May 26 2025
%H A011956 Juan E. Iglesias, <a href="https://doi.org/10.1524/zkri.2006.221.4.237">Enumeration of closest-packings by the space group: a simple approach</a>, Z. Krist. 221 (2006) 237-245. See Table 4.
%H A011956 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).
%H A011956 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/BarlowPacking.html">Barlow Packing</a>, on MathWorld-A Wolfram Web Resource.
%H A011956 Wikipedia, <a href="https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres">Close-packing of equal spheres</a>.
%F A011956 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
%t A011956 fa[p_, q_] := fa[p, q] = (p+q-1)!/(p!q!) - Sum[fa[p/d, q/d]/d, {d, Rest[Intersection@@(Divisors/@{p, q})]}];
%t A011956 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})]}];
%t A011956 rh[n_] := Sum[fa[n-q, q]+fb[n-q, q], {q, Select[Range[n/2], !Divisible[n-2#, 3]&]}] / 2; (* A371992 *)
%t A011956 fSO[n_] := Sum[fb[2n+1-q,q], {q, Select[Range[n+1,2n], !Divisible[2n+1-2#,3]&]}];(*A011954*)
%t A011956 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)*)
%t A011956 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*)
%t A011956 a[n_] := rh[n] - If[OddQ@n, fSO[(n-1)/2], fO[n/2]+fO[n/2-1]];
%t A011956 Table[a[n],{n,7,50}] (* _Andrei Zabolotskii_, May 30 2025 *)
%o A011956 (PARI)
%o A011956 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
%o A011956 (Python)
%o A011956 def A011956(n): return A371992(n) - (A011954(n//2) if n&1 else A011955(n//2)+A011955(n//2-1))
%o A011956 # _M. F. Hasler_, May 27 2025
%Y A011956 Cf. A011768, A011954, A011955, A371992.
%K A011956 nonn,easy
%O A011956 7,2
%A A011956 _N. J. A. Sloane_
%E A011956 Name and offset corrected by _Andrei Zabolotskii_, Feb 14 2024
%E A011956 Name changed by _M. F. Hasler_, May 26 2025
%E A011956 Terms a(17) onwards from _Andrei Zabolotskii_, May 30 2025