A119869 Sizes of successive clusters in f.c.c. lattice centered at a lattice point.

1, 13, 19, 43, 55, 79, 87, 135, 141, 177, 201, 225, 249, 321, 321, 369, 381, 429, 459, 531, 555, 603, 627, 675, 683, 767, 791, 887, 935, 959, 959, 1055, 1061, 1157, 1205, 1253, 1289, 1409, 1433, 1481, 1505, 1553, 1601, 1721, 1745, 1865, 1865, 1961, 1985, 2093, 2123

Offset

0,2

References

N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..9999

Paul Bourke, Waterman Polyhedra.

Paul Bourke, On-line generator

Martin Kraus, Live Graphics3d

Mirek Majewski, Dedicated commands in MuPAD

Wouter Meeussen, ConvexHull3D package & demo-file.

Mark Newbold, Waterman Polyhedra. CCPOLY Java Applet.

Steve Waterman, Waterman Polyhedron.

Steve Waterman, Missing numbers formula

Formula

Partial sums of A004015, which has an explicit generating function.

Maple

maxd:=20001: read format: temp0:=trunc(evalf(sqrt(maxd)))+2: a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od: th2:=series(a, q, maxd): a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a, q, maxd): th4:=series(subs(q=-q, th3), q, maxd):
t1:=series((th3^3+th4^3)/2, q, maxd): t1:=series(subs(q=sqrt(q), t1), q, floor(maxd/2)): t2:=seriestolist(t1): t4:=0; for n from 1 to nops(t2) do t4:=t4+t2[n]; lprint(n-1, t4); od: # N. J. A. Sloane, Aug 09 2006

Mathematica

a[n_] := Sum[SquaresR[3, 2k], {k, 0, n}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2012, after formula *)
Accumulate[SquaresR[3, 2*Range[0, 70]]] (* Harvey P. Dale, Jun 01 2015 *)

Crossrefs

Cf. A004015, A004215, A055039.

Cf. A055039 [missing polyhedra]. Properties of Waterman polyhedra: A119870 [vertices], A119871 [faces], A119872 [edges], A119873 [volume]. Waterman polyhedra with different centers: A119874, A119875, A119876, A119877, A119878.

Sequence in context: A216101 A096455 A124199 * A272200 A106904 A106903

Adjacent sequences: A119866 A119867 A119868 * A119870 A119871 A119872

Keyword

nonn,nice

Author

Hugo Pfoertner, May 26 2006

Extensions

Edited by N. J. A. Sloane, Aug 09 2006

Additional links from Steve Waterman, Nov 26 2006

Status

approved