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

1,5

Links

Table of n, a(n) for n=1..43.

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.

Formula

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

a(n+1)/a(n) = 2 - 2/n + o(1/n). - M. F. Hasler, Jun 09 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})]}]; (*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 *)

Prog

(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

Crossrefs

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

Sequence in context: A104880 A152478 A102973 * A066372 A058519 A181065

Adjacent sequences: A371989 A371990 A371991 * A371993 A371994 A371995

Keyword

nonn

Author

R. J. Mathar, Apr 15 2024

Extensions

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

Status

approved