A119872 -id:A119872 - cp's OEIS frontend

A119870 Number of vertices of the root-n Waterman polyhedron.

12, 6, 24, 12, 24, 32, 48, 54, 36, 24, 48, 24, 72, 72, 48, 60, 48, 54, 72, 72, 72, 72, 48, 56, 132, 96, 120, 96, 72, 72, 96, 102, 96, 96, 120, 84, 120, 144, 96, 72, 120, 72, 168, 168, 120, 120, 144, 168, 108, 126, 168, 72, 144, 152, 144, 144, 192, 120, 144, 144

Offset: 1

Hugo Pfoertner, May 26 2006

nonn

The root-n Waterman polyhedron is the convex hull of the intersection of a closed ball of radius sqrt(2*n) with the lattice of sphere-center points of a cubic close packing. [Probably the f.c.c. lattice is intended here. - N. J. A. Sloane, Aug 09 2006]

The basic sphere center series of Waterman polyhedra is obtained by choosing a sphere center as the center of the closed ball. Other choices are possible. An example is given in A119874 ... A119878. For n in A055039 no lattice points are hit; the corresponding polyhedra are the same as for n-1.

Cf. A119870, A119875 [vertices of void-centered Waterman polyhedron].

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

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

Original entry on oeis.org

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

Hugo Pfoertner, May 26 2006

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.

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

Cf. A004015, A004215, A055039.

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

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 *)

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

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

Additional links from Steve Waterman, Nov 26 2006

A119877 Number of edges of the root-n Waterman polyhedron of void-center type as defined in A119870.

Original entry on oeis.org

12, 12, 36, 36, 84, 48, 36, 36, 96, 108, 72, 72, 60, 156, 120, 120, 84, 72, 144, 144, 228, 228, 120, 120, 120, 156, 144, 144, 216, 120, 252, 252, 144, 264, 168, 168, 204, 192, 288, 288, 240, 192, 276, 276, 312, 288, 168, 168, 192, 252, 360, 360, 336, 288, 192

Offset: 0

Hugo Pfoertner, Jun 05 2006

nonn

For more information see A119870.

Cf. A119869, A119874, A119872 [edges of Waterman polyhedron of basic type].

cp's OEIS Frontend

A119870 Number of vertices of the root-n Waterman polyhedron.

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

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A119877 Number of edges of the root-n Waterman polyhedron of void-center type as defined in A119870.

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