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

Author

R. J. Mathar, Apr 15 2024

Keywords

• nonn

Links

• 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.

Crossrefs

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

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})]}]; (*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 *)

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

Formula

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

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

Extensions

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