This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A371992 #20 Jun 12 2025 13:15:23
%S A371992 0,0,1,1,2,3,5,8,15,23,41,70,126,223,406,740,1370,2545,4769,8977,
%T A371992 16985,32261,61469,117488,225060,432159,831235,1601796,3090926,
%U A371992 5973198,11556533,22385600,43405353,84247085,163661488,318209920,619181766,1205733457,2349558582,4581555964,8939468450,17453081143,34094082857
%N A371992 Number of different closest packings of equal spheres for rhombohedral crystals having repeat period n.
%H A371992 J. E. Iglesias, <a href="https://doi.org/10.1524/zkri.1981.155.1-2.121">A formula for the number of closest packings of equal spheres having a given repeat period</a>, Z. Krist. 155 (1981) 121-127, Table 1.
%F A371992 a(n) + A371991(n) = A000046(n).
%F A371992 a(n+1)/a(n) = 2 - 2/n + o(1/n). - _M. F. Hasler_, Jun 09 2025
%t A371992 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)*)
%t A371992 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)*)
%t A371992 am[p_] := am[p] = 2^(p-1) - Sum[am[p/d], {d, Rest@Divisors@p}]; (*A000740; Eq. (6)*)
%t A371992 atf[p_] := atf[p] = 2^(p-1)/p - Sum[atf[p/d]/d, {d, Select[Rest@Divisors@p, OddQ]}];(*A000048; Eq. (9)*)
%t A371992 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) *)
%t A371992 Table[a[n], {n, 2, 40}] (* _Andrei Zabolotskii_, May 30 2025 *)
%o A371992 (PARI) 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
%Y A371992 Cf. A371991, A000046, A051168, A180424, A000740, A000048.
%K A371992 nonn
%O A371992 1,5
%A A371992 _R. J. Mathar_, Apr 15 2024
%E A371992 Offset changed to 1 and a(1) = 0 prefixed by _M. F. Hasler_, Jun 05 2025